![E D I T O R I A L C O M M I T T E E
A M S S u b c o m m i t t e e
Robert D . MacPherso n Grigori i A . Marguli s Jame s D . Stashef f (Chair )
A S L S u b c o m m i t t e e Steffe n Lemp p (Chair )
I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair )
H . I I . >Kejio6eHK O
O C H O B H b l E C T P Y K T Y P b l H M E T O H b l
T E O P M M r i P E H C T A B J I E H M M
MIIHMO, MOCKBA , 200 4
This work wa s originally publishe d i n Russian b y MIIHM O unde r th e title "OcHOBHbi e
CTpyKTypu TeopH H npe,a;cTaBJieHHM " ©2004 . Th e presen t translatio n wa s create d unde r
license fo r th e America n Mathematica l Societ y an d i s published b y permission .
Translated fro m th e Russia n b y Ale x Martsinkovsk y
2000 Mathematics Subject Classification. Primar y 20-01 , 20Cxx ;
Secondary 1 7B1 0 , 20G05 , 20G42 .
For additiona l informatio n an d update s o n thi s book , visi t
www.ams.org/bookpages/mmono-228
Library o f Congres s Cataloging-in-Publicatio n Dat a
Zhelobenko, D . P. (Dmitri i Petrovich )
[Osnovnye struktur y i metody teori i predstavlenii . English ]
Principal structure s an d method s o f representatio n theor y / D . Zhelobenk o ; translate d b y
Alex Martsinkovsky .
p. cm . — (Translation s o f mathematical monograph s ; v. 228)
"Originally publishe d i n Russian b y MTSNMO unde r th e title 'Osnovny e struktur y i metod y
teorii predstavlenii ' c2004"—T.p . verso .
Includes bibliographica l reference s an d index.
ISBN 0-821 8-3731 - 1 (alk . paper)
1. Representation s o f groups. 2 . Representations o f algebras. I . Title. II . Series.
QA176.Z5413 200 4
512/
.22—dc22 200505235 2
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![x PREFAC E
(and uniqueness ) o f invarian t measure s o n suc h groups . I n tha t sense , th e classi -
cal Fourie r analysi s (Fourie r serie s an d integrals ) i s subsume d int o a n impressiv e
development progra m o f harmonic analysi s o n topological groups .
Logical foundations o f Fourier analysi s ca n be significantly clarifie d withi n th e
framework o f "abstrac t harmoni c analysis" , wher e th e grou p G i s replace d b y a
C*-algebra. Fundamenta l result s i n tha t directio n ar e du e t o I . M . Gelfan d an d
M. A. Naimark (i n the 1 940s) . Beginnin g with the 1 950s , the theory of C*-algebra s
develops ver y rapidl y and , t o a larg e extent , characterize s th e functiona l analysi s
of th e 20t h century . I t i s importan t t o observ e tha t tha t theor y ha s fundamenta l
applications t o operato r algebras , Hop f algebras , dynamica l systems , statistica l
mechanics, quantu m field theory, etc .
Modern representatio n theor y deal s with a wide variety of associative algebras ,
including structure algebra s of manifolds an d Lie groups, universal enveloping alge-
bras of Lie algebras, group (convolution ) algebras , Hopf algebras, quantum groups ,
etc. Notic e that th e theory of Lie groups, born within the context o f differential ge -
ometry, is now included in the framework o f functional analysi s by way of bialgebras
and forma l group s associate d wit h Li e groups.
One ma y als o expand th e definitio n o f representation theor y t o include , i f de-
sired, suc h neighborin g discipline s a s abstract theor y o f differential equations , the -
ory o f sheaves o n homogenous spaces , microanalysis, quantu m field theory , etc .
There i s a know n thesi s accordin g t o whic h "mathematic s i s representatio n
theory". Th e correspondin g antithesi s ca n b e state d a s "mathematic s doe s no t
reduce to representation theory". I t is worthwhile to note the nature of the question.
Whatever i s true , i t appear s tha t th e scop e o f representatio n theor y i s alread y
comparable wit h tha t o f the entir e mathematics .
It ma y b e that th e desir e to systematize mathematic s i n the spiri t o f represen-
tation theor y mad e N . Bourbak i writ e th e multi-volum e se t "Element s o f mathe -
matics" . Despit e certai n shortcoming s o f tha t titani c wor k (excessiv e formalism ,
unfinished parts ) on e finds original treatment o f several fundamental issues , includ-
ing general aspects of algebra, topology , the theory of integration, th e theory of Lie
groups an d Li e algebras, etc .
At present, ther e is a large number o f monographs dealin g with various aspect s
of representation theory , includin g Lie groups and Lie algebras ([4] , [10], [14], [31],
[35], [61 ]) . Banac h algebra s ([6] , [8] , [1 3] , [22] , [49] , [58]) , algebrai c group s ([3] ,
[29], [64] , [73]) , infinite-dimensiona l group s ([53]) , genera l representatio n theor y
([40]). Th e author' s monograp h [75 ] ca n b e use d a s a n easil y accessibl e sourc e
of informatio n o n representation s o f Li e groups , especiall y suitabl e fo r physicists .
However, ther e i s still no monograph whic h would put togethe r al l of those aspect s
of representation theory .
To fill th e gap , thi s boo k wa s conceive d a s a compilatio n o f canonica l text s
on representatio n theory . I t provide s a systemati c descriptio n o f a wid e spectru m
of algebro-topologica l structures . O n on e hand , th e concep t o f suc h a boo k i s
appealing becaus e i t allow s u s t o compar e idea s an d method s fro m differen t part s
of representatio n theory . O n th e othe r hand , i t i s als o risk y jus t becaus e o f th e
sheer volum e o f th e materia l t o b e covered . Nevertheless , th e autho r think s tha t
a partial resolutio n o f this dilemm a i s possible because th e offere d text s hav e bee n
carefully worke d upo n an d refined .](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-15-320.jpg)
![PREFACE x i
The content s o f the boo k spli t int o three parts . Par t I (Introduction ) contain s
general fact s fo r beginners , includin g linea r algebr a an d functiona l analysis . Th e
survey-type section s o n topology , theor y o f integration , etc . (se e [23] , [24] , [26] ,
[31]) a s wel l a s Appendice s A , B , C , an d D ar e writte n i n th e sam e spirit . I n
the mai n Par t I I (Genera l theory ) w e conside r associativ e algebras , Li e algebras ,
topological groups , an d Li e groups. W e als o mentio n som e aspect s o f rin g theor y
and th e theor y o f algebrai c groups . W e provid e a detaile d accoun t o f classica l
results i n those branche s o f mathematics, includin g invarian t integratio n an d Lie' s
theory o f connection s betwee n Li e group s an d Li e algebras . I n Par t II I (Specia l
topics) w e conside r semisimpl e Li e algebr a an d Li e groups , Banac h algebras , an d
quantum groups .
The boo k bring s th e reade r clos e t o th e moder n aspect s o f "noncommutativ e
analysis", includin g harmoni c analysi s o n locall y compac t groups . Th e autho r
regards th e content s o f this book a s a prerequisite fo r thos e who want t o seriousl y
study representatio n theory .
The styl e o f the boo k allow s the autho r t o choos e th e dept h o f the expositio n
to his taste. Fo r example, we prove the theorem o n the conjugac y o f Cart an subal -
gebras (i n complex Lie algebras) bu t omi t a similar resul t fo r Bore l subalgebras (i n
semisimple Li e algebras) . Ye t th e autho r hope s tha t th e reade r wil l see a detaile d
enough panorami c descriptio n o f representation theory .
The diverse nature of the compiled materia l unavoidably lead s to discrepancie s
in traditions , whic h sometime s caus e certai n redundanc y i n th e definition s an d
notation. Fo r example , th e notatio n End X i n th e categor y o f vecto r space s i s
sometimes replace d b y L(X) wher e dimX < oo .
The exercises included in the book, as a rule, are designed as tests for beginners.
Sometimes (i n moderation ) th e result s o f the exercise s ar e use d t o shorte n certai n
proofs. Onl y th e exercise s marke d wit h a n asteris k ca n b e viewe d a s mor e o r les s
serious problems .
While workin g o n th e boo k th e autho r fel t himsel f a chronicler . Indeed , th e
book cover s a century i n the developmen t o f mathematics, a period whic h i s prob-
ably no t ye t full y appreciated .
The content s o f the boo k are , t o a larg e extent , base d o n tw o electiv e course s
the author gave at the Independent Universit y of Moscow in 1996-1998. Th e lectur e
notes of one of those courses were published i n 2001 ([78]). Th e work on this boo k
was partially supporte d b y the RFF I Gran t 01 -01 -0049 0 an d NW O 047-008-009 .
The author i s grateful t o V. R. Nigmatullin for his help during the proofreadin g
of the text .
D. Zhelobenk o](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-16-320.jpg)
![4 1. BASI C NOTION S
with respec t t o th e compositio n o f endomorphism s
(1.3) (ab)x = a(bx)
where a,b E EndM, x G M. I n thi s example , th e identit y transformatio n ex = x
(x G M) i s the identity .
2. Th e subse t Aut M c EndM , consistin g o f automorphisms (i.e. , invertibl e
endomorphisms) o f M, i s a group (calle d th e automorphism group of M).
The associativit y axio m (1 .1 ) ca n b e viewe d a s a n abstractio n o f th e corre -
sponding propert y o f EndM .
Examples of abelian groups include: Z (the set of integers), Zp (th e cyclic group
of integer s modul o p) , Q (th e se t o f rationa l numbers) . A s example s o f abelia n
semigroups we mention Z + (th e set of nonnegative integers ) an d N = Z + {0} (th e
natural numbers) .
1.2. Ring s an d fields . A set R wit h tw o associativ e binar y operations , ad-
dition (x , y) -+ x + y an d multiplication (x, y)i
— > xy, i s called a ring i f R i s a grou p
with respec t t o additio n an d th e distributivity condition s hold :
(1.4) x(y + z) — xy + xz, (x + y)z = xz + yz
for al l x,y,z G R. I n particular , R ha s a zero element 0 such tha t 0 • x = x • 0 = 0
for eac h x G R.
The rin g R i s said t o b e commutative (respectively , a ring with identity) i f R
is commutativ e (respectively , ha s a n identity ) wit h respec t t o multiplication . A
ring wit h identit y i s calle d a skew field i f ever y nonzer o elemen t i s invertible . A
commutative ske w field i s called a field.
A subse t G C G (respectively , R C R) i s calle d a subgroup of th e grou p G
(respectively, a subring o f R) i f G i s close d unde r th e grou p operation s (x,y) i
— »
xy~1
(respectively , i f R i s close d unde r th e rin g operations) . I n tha t cas e G
(respectively, R) i s called a n extension o f G (respectively , o f R).
The identit y elemen t o f the rin g R i s usually denote d 1 . W e shall als o assum e
that a ring wit h identit y i s nontrivial, i.e. , 0 ^ 1 .
Examples. 1 . I f R i s a ring wit h identity , the n R contain s th e subrin g R o f
multiples o f the identity , i.e. , 0, ±n, wher e n = 1 + • • • + 1 (n summands) .
2. An y commutativ e rin g R give s ris e t o a n extensio n R[x], consistin g o f al l
polynomials i n x. Here , by a polynomial w e understand a n element o f the for m
f{x) = a0 + ax H h a nxn
,
with coefficient s ai G R (i — 0,1,..., n). A mor e detaile d analysi s o f thi s notio n
will be given i n 4.1.
We also recall the standar d notatio n R , C ,HI (respectively) fo r th e field of real
numbers, th e field of complex numbers , an d th e ske w field of real quaternions .
1.3. Vecto r spaces . Give n a field F , a se t X wit h operation s o f additio n
X x X— > X, (x , y)H ^ x -f- y an d multiplicatio n F x X— > X, (A , x)i- » Ax, is called
a vector space over F i f X i s an abelia n grou p with respec t t o additio n an d
(1.5) {x + y) = x + Ay,( A + ji)x = x + fix
for al l A , i G F an d x , y G X. I n particular , 0 • x = 0 for eac h x G X, wher e 0 in
the left-han d (right-hand ) sid e i s the zer o elemen t o f F {X). Th e element s x G X
are calle d vectors, an d th e element s A G F scalars. W e shall cal l the operation s of
addition an d multiplicatio n th e vector operations o n X.](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-21-320.jpg)
![8 1. BASI C NOTION S
We shal l occasionall y us e th e categorica l vocabular y t o shorte n som e state -
ments. Sometime s (i n Part III ) suc h shortenings wil l be substantial .
Example. Fo r eac h morphis m a: M — » N i n SE T w e ca n defin e th e dua l
morphism a * : F(N) - • F(M) (se e 1 .3 ) b y
(1.12) (a*f)(x) = f(ax),
where / G F(N) an d x G M. I t i s eas y t o chec k tha t (ab)* = 6*a * fo r composi -
tions (1 .1 1 ) .
Thus th e ma p Mi
— > F(M) give s ris e to a contravarian t functo r fro m SE T t o
the categor y o f algebras ove r F.
2. Vecto r space s
2.1. Notation . Le t X b e a vecto r spac e ove r a field F. W e shal l conside r
systems of elements e = (e^)^ / o f X (wher e / i s an arbitrar y se t o f indices). Finit e
sums
(2.1) x = J2 X
^
whereA^ G F (wit h only finitely man y A ; ^ 0 ) are called linear combinations o f the
ei (i e I). Th e se t Fe o f al l suc h linea r combination s i s called th e linear hull (or
span) o f e.
Clearly, Fe i s a subspac e o f X. I f X — Fe, the n w e shal l sa y tha t X is
spanned b y e .
A system e is said t o b e linearly independent (respectively , spanning) i f x = 0
in (2.1 ) implie s tha t A ; = 0 for eac h i G I (respectively , i f F e = X). Linea r inde -
pendence i s equivalent t o the uniquenes s o f decomposition (2.1 ) . I n that cas e (2.1 )
is written a s
yz.z) x — y x^e^,
i
where th e Xi (i G I) ar e uniquel y determine d b y x. Her e w e set Xi — Si(x). Th e
coefficients xi (i G I) ar e calle d th e coordinates o f x (wit h respec t t o e) .
Each linearl y independen t spannin g syste m e in X i s called a basis of X. An y
vector x G X i s the n uniquel y represente d i n th e for m (2.2) , wit h coordinate s
%i ~ &i %) •
Examples. 1 . Th e coordinat e spac e Fn
(se e 1 .3 ) ha s a basis consisting o f th e
elementse ^ = (0,... , 0,1, 0,..., 0) wit h 1 at it h place . I n thi s cas e th e notatio n
x — (xi,..., xn) identifie s th e coordinate s X{ — £i(x) o f x G Fn
wit h respect t o th e
basis (ei), wher e i — 1,..., n.
2. Le t F[M] C F(M) b e the subse t o f all finite function s f:M->F, i.e. , th e
functions whic h ar e no t zer o only a t a finite numbe r o f points o f M. Clearl y F[M]
is a vecto r subspac e o f F(M) wit h "delta-functions " S a (a G M) a s a basis, wher e
da(x) = 0 when x ^ a and S a(a) = 1 .
Henceforth w e shall identif y a G M wit h th e delta-functio n 5 a G F[M. A s a
result, M embed s (a s a basis) i n F[M. Th e space F[M] i s called th e formal linear
hull (ove r F) o f M.](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-25-320.jpg)
![2. VECTO R SPACE S 9
2.2. Bases . Accordin g to the well-known theorem of Hamel (see , for example ,
[1]), every vector space X over a field F has a basis. Th e proo f o f this theore m i s
based o n the Zor n lemma , whic h implie s th e existenc e o f a maximal (wit h respec t
to inclusion ) linearl y independen t syste m i n X. A s a n eas y exercise , th e reade r
should chec k that eac h such syste m i s a basis of X.
A substantial refinemen t o f Hamel's theore m i s that any two bases of X are of
the same cardinality. W e shall sketch a proof o f this statement .
If X admit s a finite basis, then th e proof i s an easy exercise. Assum e now tha t
X ha s base s A an d B wit h infinit e cardina l number s a = card A an d (3 — card B.
Note that A is a disjoint unio n of subsets An ( n G N), where An consist s of elements
a G A whic h ca n be represented a s a linear combination , wit h nonzer o coefficients ,
of exactly n element s b G B. I t i s easy t o chec k that car d .An < nf3 n
. Therefor e
oo
(2.3) a < ]Tn/T.
7 1 = 1
It i s well known (see , for example , [65] ) that (3 2
— (3 for eac h infinite cardina l f3, so
that j3 n
— (3 for eac h n G N. Moreover , th e right-han d sid e of (2.3 ) coincide s wit h
j3 and therefor e a < (3. Similarly, (3 < a an d therefor e a — j3.
Thus every vector space X give s rise to a unique cardinal number dimX , calle d
the dimension o f X, an d define d a s the cardinalit y o f an arbitrar y basi s of X. Th e
space X i s said to he finite-dimensional (respectively , infinite-dimensional) i f dim X
is finite (respectively , infinite) .
Examples. 1 . dimF n
= n.
2. Ther e ar e vecto r space s o f arbitrar y dimension . Fo r example , dimF[M ] =
cardM.
Exercise. Chec k (similarly to the proof of Hamel's theorem) that ever y linearly
independent syste m i n a vecto r spac e X i s containe d i n som e basi s o f X. Fo r
example, eac h nonzer o vector i s part o f a basis of X.
2.3. Proposition . Let a G Horn (X, Y) (znVECT F). Then
(a) For each basis e^ (i G /) of X, the operator a is uniquely determined by its
values aci.
(/?) For each collection f % G Y (i G I), there is an operator a G Horn (X, Y)
such that aei — fi for all i G /.
PROOF. Fo r the first assertion, it suffices t o notice that, in the notation of (2.2),
(2.4) ax — y^ jXja(el).
i
For th e secon d assertion , on e quickl y check s tha t th e sam e formul a define s a as a
linear map . •
As a consequence, w e have the rul e
(2.5) H o m ( X , Y ) ^ y a
,
where a = dimX .
2.4. Corollary . The equality dim X = dim F for X and Y in VECT^ ? is
equivalent to X « Y.](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-26-320.jpg)
![10 1. BASI C NOTION S
Indeed, i f d i m X = di m Y, the n th e formul a ae % — fi, wher e a an d fa ar e base s
(respectively) i n X an d Y , define s th e desire d isomorphis m X « Y .
Examples.1 . Eac h n-dimensiona l vecto r spac e X (n G Z+ ) i s isomorphi c t o
Fn
.
2. Eac h vecto r spac e X ove r a field F i s isomorphi c t o F[M] , wher e d i m X =
cardM.
2.5. Matrices . Le t e = (cj) jej an d / = {fi)iei b e base s i n vecto r space s X
and (respectively ) Y . Expressin g ae 3 (j G J) i n term s o f th e element s o f / , w e hav e
(2.6) ae 3 = ^ P a ^ / i *
i
with a tJ G F.
The collectio n a/ 5e = (a^) , wher e i G / an d j G J, i s calle d th e matrix o f a (i n
the base s / an d e) . Th e element s a ^ wit h fixed i (respectively , j) ar e calle d th e
rows (respectively , columns) o f th e matri x a/ }€>. Accordin g t o (2.6) , th e column s
aej o f aj 5 e satisf y th e followin g finiteness condition :
($) Fo r eac/ i j E J only finitely many numbers a^ are different from zero.
Conversely, i t follow s fro m 2. 3 (/3 ) tha t eac h matri x a^ e satisfyin g conditio n
(<£) defines a uniqu e operato r a G Horn (X, Y) wit h a/ ?e a s it s matri x (wit h respec t
to / an d e) . A s a consequence , th e ma p a -* af^ e i s a n isomorphis m betwee n
Hom(X, Y ) an d th e vecto r spac e o f al l matrice s (2.6 ) satisfyin g conditio n ($) .
If th e base s / an d e ar e fixed, w e shal l simpl y writ e a instea d o f a/ >e (i n othe r
words, w e shal l identif y th e operato r a G Hor n (X, Y) wit h it s matri x (2.6)) . A s
an exercise , th e reade r shoul d chec k tha t th e compositio n o f operator s a — be
(whenever i t i s defined ) correspond s t o th e standar d matri x multiplicatio n
(2.7) a,ij = }^b ikckj.
k
In particular , th e algebr a E n d X i s isomorphi c t o th e matri x algebr a M a t ( n , F ) ,
where n = dimX , consistin g o f al l squar e matrice s o f orde r n satisfyin g condi -
tion(<£>) .
Normally, w e shal l us e th e symbo l Ma t (n , F) whe n n < oo . I n tha t cas e
Mat (n , F) consist s o f al l n x n matrice s ove r F.
2.6. Dua l spaces . A vecto r spac e X * consistin g o f al l F- valued linea r func -
tions (als o calle d functionals ) o n X i s calle d th e (algebraically ) dual space o f X
(X* = H o m ( X , F ) ) .
As i n (2.5) , th e value s
(2.8) f(x) = Y,fi x
*>
where Xi — £i(x) ar e th e coordinate s o f x i n th e basi s e (se e 2.1 ) , o f eac h functiona l
/ G X* ar e uniquel y determine d b y th e coefficient s f t G F. I n tha t sense , (2.8 ) ca n
be writte n a s
(2.9) X * ^ F a
,
where a = d i m X .](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-27-320.jpg)
![2. VECTO R SPACE S 11
Let e — (ei)iei b e a linearl y independen t se t i n X. W e wan t t o sho w that ,
in X*, ther e exist s a dual system Si (i G I) give n by
(2.10) £i(e j) = 5 ij,
where 6ij i s Kronecker' s delt a (5 i3 — 0 when i ^ j , and Su = 1 ) . Indeed , e ca n
be extended t o a basis of X (se e 2.2). No w define Si(x) = Xi as the coordinate s of
x G X i n that basi s (se e 2.1).
In particular , fo r eac h nonzer o x G X ther e i s an / G X* suc h that f(x) ^ 0 .
The dual systems allow us to find explicit expressions for matri x elements (2.6 )
of a. Namely ,
(2.11) dij = ipi(aej),
where ipi (i G I) i s the syste m dua l t o the basi s fi (i G I)o f Y.
Examples. 1 . Th e vecto r spac e F n
ca n b e identifie d wit h it s dua l vi a (2.8) ,
where i = 1 ,.. . , n.
2. Th e spac e dua l t o F[M] ca n be identifie d wit h F(M) usin g the formul a
9(f) = £/(*)<?(*),
X
where / G F[M) an d # G F(M).
2.7. Bilinea r forms . A bilinear for m f: X xY -+ Z (in VECTF) i s said t o
be nondegenerate i f for eac h 0 / x G l (respectively , 0 ^ y G Y) ther e i s a 7 / G Y
(respectively, x G l ) suc h that /(# , y) ^ 0 .
The nondegenerac y o f / i s equivalen t t o th e injectivit y o f eac h o f th e map s
x»
—> f x an d y *-* fy, wher e
(2.12) fx(v) = f(x,y) = f y(x).
In thi s sens e (2.1 2 ) give s ris e t o embedding s X (respectively , Y ) i n Hor n (Y, Z)
(respectively, Hor n (X, Z)).
In general , th e left kernel ker ^ / (respectively , th e right kernel ker p /) o f /
is define d a s th e kerne l o f th e ma p x i
— > fx (respectively , y -^ f y). Thu s / i s
nondegenerate mean s that ker> , / = ker p / = 0 .
Examples.1 . I f Z = F an d / i s nondegenerate , the n (2.1 2 ) give s ris e t o
embeddings X - • Y * an d Y -> X* .
2. Th e bilinea r for m (x,?/ ) = y(x ) i s nondegenerate o n X x X * (se e 2.6) . I n
this cas e (2.1 2 ) become s
(2.13) x(y) = (x,y)=y(x),
so that th e first equalit y give s ris e t o a n embeddin g o f X int o th e bidua l X* * =
(X*)*.
3. I f dimX < oo , then thi s embeddin g i s an isomorphism , X w X**.
4. Formul a (2.1 1 ) ca n als o be written a s
(2.14) dij = (aej,(pi).
The bilinea r for m (2.1 3 ) i s called th e canonical bilinear form o n X x X* .
Exercise. Le t (• , •) be a nondegenerat e symmetri c bilinea r for m o n a finite-
dimensional vecto r spac e X. Th e relation s
x(y) = (x,y) = y{x)](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-28-320.jpg)
![2. VECTO R SPACE S 13
An operato r a G En d A i s sai d t o b e block-diagonal (relativ e t o decomposi -
tion (2.1 9) ) i f dij = 0 whe n i ^ j . I n thi s cas e w e writ e a = 0 z a % or a = dia g (a^) ,
where ai — a^. I f / = { 1 , . . . , n} , then w e writ e
n
(2.22) a = ££)a l = d i a g ( a i , . . . , a n ) .
Exercise. Linea r functiona l / G A* o n (2.1 9 ) ar e o f th e for m
(2.23) f(x) = Y^fi(xi),
i
where th e f t G A* (i G /) coul d b e arbitrar y linea r functionals . Therefor e
(2.24) A * = []X *
i
Note tha t (2.23 ) i s a finit e su m (sinc e an y x G A i s finite) .
2.10. Quotien t spaces . Le t A Q be a subspac e o f a vecto r spac e A . Settin g
x ~ y whe n x — y G AQ , we hav e a n equivalenc e relatio n o n A wit h equivalenc e
classes
(2.25) IT(X)=X + X Q,
where x G A (i.e. , 7r(x ) i s th e se t o f al l vector s x + y, wher e y G A0 ). Th e relatio n
TT(X) = n(y) i s equivalen t to x ^ y. I t i s no w clea r tha t th e quantitie s define d b y
(2.26) XTT(X) — 7r(Ax), 7r(:r ) + 7r(y ) = 7r( x + y)
do no t depen d o n th e choice s o f x an d y i n th e classe s 7r(x ) an d rc(y). Th e ne w
vector spac e 7r(A ) i s denote d A/A o an d calle d th e quotient space o f A b y th e
subspace XQ.
Note tha t (2.26 ) mean s tha t ix X —> X / A Q i s linear . Th e ma p TT i s calle d th e
canonical projection (o f A ont o A/Ao) .
Choosing a basi s e o i n A o an d extendin g i t t o a basi s eoUe i i n A w e hav e
(2.27) A = A 0 © Ai, wher e X 1 = Fe x « A / A 0 .
Therefore di m A = di m Ao + di m X. Th e secon d summan d i s calle d th e codimen-
sion o f A Q i n A an d i s denote d codi m XQ.
Example. I f codi m AQ = 1 , the n A 0 i s a hyperplane i n A determine d b y th e
linear equatio n TT(X) — 0. Accordingly , (2.27 ) become s
(2.28) A = A 0 © F e 0 ,
for an y e o no t containe d i n AQ . I t i s als o clea r tha t classe s (2.25 ) coincid e wit h
hyperplanes TT(X) = const .
2.11. Invarian t subspaces . A subspac e A Q C A i s said t o b e invariant wit h
respect t o a G EndA " i f aA' o C A' o (i.e . ax G A^Q for eac h x G Ao). I n tha t cas e th e
relation
(2.29) 7r(a)7r(x ) = 7r(ax)
gives ris e t o a n induced operator a = re {a) G End A, wher e A = A / A 0 .
If w e represen t operator s a G En d A a s bloc k matrice s a — (a 2J), wher e
z,j = 1 , 2 (i n accordanc e wit h (2.27)) , the n th e invarianc e conditio n aA 0 C A 0
is equivalen t t o th e conditio n a^ = 0 (i.e. , t o a bein g uppe r triangular) .](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-30-320.jpg)
![14 1. BASI C NOTION S
Exercise. Eac h operato r a G Horn (X, Y) give s rise t o a vector spac e isomor -
phism
(2.30) X/kera^ima.
2.12. Th e groun d field . Late r i n thi s boo k w e will hav e t o impos e certai n
conditions o n the groun d field F.
The field F is said to b e algebraically closed if F contains th e root s o f all alge-
braic equations f(x) — 0, where / i s a polynomial with coefficients i n F. I n general,
F has an algebraic closure, i.e., an algebraically closed extension F (meaning that F
is embedded i n F as a subfield). See , for example , [1 ] . Her e is a classical example :
C i s an algebrai c closur e of R .
We say that F is a field of characteristic p — 0 (respectively, p ^0 ) if n ^ 0 for
each n G N (respectively , p is the smalles t o f the number s n G N such tha t n — 0).
Here, by definition, n = H h i (see 1.2). Th e characteristic of F i s denoted
p = charF. I f p / 0 , then p is a prime numbe r (exercise) . Her e is an example : th e
residue field Fp = Z/pZ.
Sometimes we shall consider generalizations of vector spaces (and, in particular,
of algebras) wher e th e groun d field F is replaced wit h a ring R.
For example, the matrix algebr a Mat (n, R) ove r a ring R, wher e n is a cardinal
number, i s defined a s the algebr a o f al l matrice s wit h element s a^ G R(z, jG J),
where card / = n, satisfyin g conditio n ($ ) fro m 2.5 .
Exercises. 1 . Prove tha t Hamel' s theore m generalize s t o vecto r space s ove r
skew fields.
2*. Tr y to prove the Frobenius theorem: eac h finite-dimensional ske w field over
R i s isomorphic t o on e of the ske w fields R, C, H.
3. I n particular , eac h finite-dimensional field ove r R is isomorphic t o eithe r R
or C . Se e also [1].
3. Element s o f linea r algebr a
3.1. Determinants . Throughou t thi s sectio n X wil l b e a finite-dimensional
vector spac e ove r a n algebraicall y close d field F. A s is well known , i n this case
matrix theor y become s a n effectiv e too l fo r studyin g linea r operator s a G EndX.
First not e tha t th e notio n o f determinan t extend s fro m th e matri x a e = a ee
(see 2.5 ) t o th e operato r a G EndX. Indeed , fo r eac h pai r o f base s e and / o f X
the matrices ae an d aj are similar (se e 2.8), whence detae = det ay. Therefore , th e
equality
(3.1) de t a = det ae
defines a unique numbe r de t a, calle d th e determinant o f a. Th e languag e o f ma -
trices ca n b e use d t o stat e th e well-know n criterio n fo r invertibilit y i n EndX : an
operator a G EndX i s invertible if and onl y i f det a ^ 0.
An elemen t a G Fis called a n eigenvalue o f a if there exist s a nonzero vecto r
(an eigenvector) x G Xsuch tha t ax = ax, i.e. , (a — a)x = 0. Suc h a vector exist s
if and onl y i f the operato r a — a is not invertible , i.e. ,
(3.2) de t (a - a) = 0.
Here we identify a G Fwith th e operato r a • 1 G EndX ( 1 is the identit y operato r
in EndX) . Sinc e F is algebraically close d an d th e left-han d sid e o f (3.2) i s a](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-31-320.jpg)
![3. ELEMENT S O F LINEA R ALGEBR A 17
where the left-hand side denotes the result of the substitution A n a in p a.
Indeed, i f a ha s a unique eigenvalu e a , the n Theore m 3. 4 show s tha t
(3.14) (a-a) n
= 0 .
In the genera l case , Theorem 3. 4 implie s tha t
k
i=
where mi — dimX^, whic h yields (3.1 3) .
Exercise. Sho w that fo r a given operator a G End X th e followin g condition s
are equivalent :
(a) (a — a)n
= 0, where n = dimX ;
(/?) fo r eac h x e X ther e i s a k — k(x) suc h that ( a — a)k
x — 0;
(7) a ha s a unique eigenvalu e a ;
(5) th e characteristi c polynomia l pa i s of the for m (3.7) .
3.6. Roo t subspaces . Theore m 3. 4 would b e mor e complet e i f we explicitl y
described th e subspace s Xi (i = 1 ,... , k).
For eac h A G F le t X(a) b e the se t o f all vectors x e X satisfyin g
(3.15) (a-X) k
x = 0 whe n k>k 0(x).
Obviously X(a) i s an a-invarian t subspac e o f X. I t i s called th e root subspace of
a corresponding t o th e root A.
It i s clea r tha t X(a) contain s th e eigenspace ker( a — A) of a spanne d b y th e
eigenvectors o f a correspondin g t o th e eigenvalu e A . Moreover , X(a) 7 ^ 0 onl y
when ker( a — A)7^ 0, i.e., when A i s an eigenvalue o f a. I t i s also clear tha t a has a
unique eigenvalu e A i n X(a).
The subspac e X 0(a) i s called th e nil-space o f a.
3.7. Theorem . X is a direct sum of the root subspaces of a :
(3.16) X = 0XA (a),
A
where X(a) 7 ^ 0 only when A = A ^ (i = l,...,n ) in the notation of 3.4 , so
that (3.1 6 ) coincides with (3.8) .
PROOF. A S wa s show n i n 3.4 , eac h a G EndX i s a direc t su m a$ © a', wher e
ao is nilpotent an d a' i s invertible. Applyin g thi s t o th e operato r a — A, where A is
an eigenvalu e o f a, w e have
X = X A (a)©X/
,
where X{a) 7 ^ 0 and X' i s an a-invarian t subspac e i n which A is not a n eigenvalu e
of a . Th e desire d resul t ca n no w b e obtaine d b y inductio n o n dim X applie d t o
X'. ^ •
Exercise. Verif y directly , using definition (3.1 5) , that th e subspaces X(a) ar e
linearly independent . [Hint : Inductio n o n the lengt h o f the relation s
xi- - xm = 0,
where Xi G Xz(a) an dA ^7^ Xj whe n i 7 ^ j. Fo r a suitabl e n , appl y th e operato r
(a — Ai)n
t o suc h a relation an d the n appl y ( a — Ai)_n
.]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-34-320.jpg)
![20 1. BASI C NOTION S
Similarly, th e differenc e r = p — q i s a projectio n operato r i f an d onl y i f p an d
q ar e relate d b y th e order relation: q < p: pq = qp = g , whic h i s equivalen t t o
pX = qX © rX, i.e. , qX C pX. 1
A famil y o f projection operator s (pi)iei i s said t o b e orthogonal i f the secon d
condition of (3.20) is satisfied. A family (pi)iei i s called an orthogonal decomposition
of identity i f both condition s (3.20 ) ar e satisfied. Clearly , in that cas e X i s a direc t
sum o f the subspace s Xi = piX.
Exercises.1 . A n operato r a G EndX i s diagonal , a = dia g («i,... , an ), i n
the basi s e — (ei,..., en) i f and onl y if
n
(3.21) a = YlajPi,
where Pi is the projectio n o f X ont o Xi = Fe{ (i = 1 ,... , n).
2. I f oti 7^ aj whe n i ^ j , the n th e operator s pi ca n b e written a s
a — a*
(3.22) Pl =n
(%i — a n-
3. Sho w tha t eve n i n th e genera l case , th e projectio n operator s pi : X — >A Q
in (3.8) are polynomials in the operator a. [Hint : Us e the operators pik — {a—i) k
qi,
where
(3.23) Qi = [J(a, -aj) n
'"
with exponent s n ^ = dimX ^ ( 0 < k < rii — 1 ). ]
4. Functiona l calculu s
4.1. Th e algebr a F[x. Le t F[x ] b e a vector spac e ove r a field F wit h basi s
the symbol s x k
, wher e k G Z+. Element s / G F[x] ca n b e uniquel y writte n a s
(4-1) /(* ) = ! >xf c
,
fc=0
with /fc G F, an d are called (formal) polynomials in the independent variable x = x
The relatio n x k
xr
= x fc+r
, wher e k,r £ Z + , uniquel y extend s (b y bilinearity )
to a multiplicatio n i n F[x] suc h tha t F[x] become s a n associativ e (commutative )
algebra wit h identit y x° — 1over F.
Polynomials (4.1 ) admit substitution s x na (i.e. , xk
-*ak
) fo r each fixed a G A,
where A i s an associativ e algebr a ove r F. Mor e precisely ,
n
(4-2) /(a ) = ]T/ fcafe
.
fc=0
The multiplicatio n rul e in F[x] shows that 7 : f(x) ^ f(a) i s an algebra homomor -
phism fro m F[x] t o A, whic h i s uniquely determine d b y th e substitutio n x1—» a.
In particular , fo r eac h A G F th e numbe r /(A ) i s called th e value of f G F[x]
at X e F. Thu s th e ma p A>
—
>• /(A) , fo r a fixed / , i s a numerica l polynomia l (o r a
polynomial functio n / : F — • F).
1
The las t tw o statement s ar e vali d onl y i f th e characteristi c o f th e bas e fiel d i s differen t
from 2 . Editor's Note.](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-37-320.jpg)
![4. FUNCTIONA L CALCULU S 21
If F is infinite, the n (a s is easily verified usin g the Vandermonde determinants )
the numerica l function s /^(A ) = k
ar e linearly independent . I n that cas e the ma p
f(x) i- > /(A) i s an isomorphism betwee n F[x] an d th e algebra P(F ) o f numerical
polynomials / : F —> F.
In general , th e map 7: f(x)
1— > f(a) (fo r a fixed a G A) is called a functional
calculus (o f class F[x] ) i n A. Similarly , an y extension o f 7 to a homomorphis m
7: $—> A, where $ is an extension of F[x], is called a functional extension of class
$ i n A.
For eac h polynomia l / G F[x ] it s derivative f G F[x ] i s defined b y the rule
[xk
)' — kxk
~1
fo r each k G Z+, i.e.,
n
(4.3) /'(* ) = Efc/fe^-1
.
fc=0
Similarly one defines th e higher-order derivatives f^ — (f<<rn
~1
">
y, wher e m G Z+
and /(° ) = /. Not e that /( n + 1
) = 0 for each polynomia l (4.1 ) .
Later w e will also need th e followin g analo g of the Taylor formula i n F[x.
4.2. Proposition. Let cha,r F — 0 andx andy be two independent (commuting)
variables. Then, for each f G F[x],
(4.4) /( * +y) = £ ^ r V ,
A;= 0
where n is the degree of polynomial (4-1 )-
PROOF. Sinc e bot h side s of (4.4) depen d linearl y o n /, i t suffices t o consider
the cas e f(x) = xn
. The n
/(/c)
(x) = n(n - 1 ) • • • (n - k + l)x n
'k
for eac h k G Z+ , an d (4.4) become s Newton' s binomia l formul a
(4.5) (z+y r = £(™y-v,
where (^ ) are the binomial coefficients . Formul a (4.5 ) is proved by induction o n n.
For thi s it is essential tha t x an d y commute. Th e assumptio n cha r F = 0 implies
that k ^ 0 for each k G Z+ (i n particular, 0 ! = 1 ) . Thu s (4.4 ) i s proved fo r al l
/ G F[z] . •
4.3. Function s o f operators. Assum e (fo r th e sak e of simplicity) tha t F =
C, an d let X b e a finite-dimensional vecto r spac e ove r C . Recal l (se e 3.8 ) that
each a G EndX ha s canonica l decompositio n (3.1 7) , where Se = e6. Th e smalles t
number m G Z+ such that £ m + 1
= 0 is called th e nilpotency height o f a = S +£.
Let m = 0 , i.e., a = 6 = dia g (c*i,... , an ) i n some basi s e in X. Fo r each
function / : F—> F w e set
(4.6) /(a) x = /(A)x whe n x G XA(a),
i.e., /(a) = diag (/(ai),... , f(a n)) i n some basis e. Thi s definition doe s not depen d
on the choic e of the basi s e in X.](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-38-320.jpg)
![24 1. BASI C NOTION S
X i s called a normed space if a norm p(x) = ||x| | i s defined o n it. Accordingly ,
metric (4.1 9 ) i s given on X. A normed spac e X i s a Banach space if X i s complete
with respec t t o metri c (4.1 9) .
Exercises.1 . Prov e tha t th e vecto r operation s o n a norme d spac e X ar e
continuous (wit h respec t t o metri c (4.1 9)) .
2. Prov e tha t eac h seminor m p on X satisfie s
(4.20) p(x) - p(y) < P(x - y)
for al l x,y G X.
3. Prov e (usin g (4.20) ) tha t th e functio n p(x) = x is continuou s (o n th e
normed spac e X).
Examples. 1 . Th e spac e X — Fn
i s Banach wit h respec t t o each nor m
^ i / p
(4-21) INI P = { E W P
)
2 = 1
where 1 < p < oo .
2. Le t lp be the set of all sequences x — (#1 ,... , xi,...), wit h X{ £ -F , satisfying
the conditio n
oo
(4.22) J2^ P <OG
-
i=l
The spac e X = Z p is Banach wit h respec t t o nor m (4.21 ) , assuming n = oo .
3. Le t C(a , 6) be the space of all continuous (F- valued) function s o n [a , b]. The
formula
(4.23) H/l l = ma x |/(i) |
a<t<b
defines th e uniform norm o n C(a,b), wit h respec t t o whic h C(a,b) i s a Banac h
space.
4. Le t T b e a se t o f Lebesgu e measur e /i , an d L p(T,n) th e vecto r spac e o f
measurable function s / : T — > F suc h that |/| p
i s integrable. The n th e formul a
(4-24) H/llp={/l/(t)l P
dM(<)} "
defines a seminorm o n L v(T,ji). I t take s valu e zer o on th e function s / ~ 0 , wher e
f ~ g means tha t / an d g are equal almos t everywher e (wit h respec t t o //) . Semi -
norm (4.24 ) become s a nor m i f the function s / ~ 0 are considere d t o b e zero . I n
that sens e L P(T,IJ) i s a Banach space .
The proofs o f these assertions can be found i n textbooks o n functional analysi s
(see, fo r example , [43] , [66]) . Norm s (4.21 ) an d (4.24 ) ca n b e extende d t o includ e
the valu e p — oo. Fo r example, i n case (4.21 ) w e set
(4.25) Hallo o =supjx,j .
i
Similarly, Loo(T , /j,) is define d a s th e spac e o f essentiall y bounde d (i.e. , bounde d
almost everywhere ) measurabl e function s / : T — > F. The n Loo(T,/i ) i s a Banac h
space with respec t t o th e nor m
(4.26)
1 /
1
1 00 = in f (su p |/(t)|),
T~S S
where T ~ S mean s fi(T S) = 0 .](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-41-320.jpg)
![4. FUNCTIONA L CALCULU S 25
In particular, le t T — [a, b] an d let /i be the standard Lebesgu e measure (gener -
ated b y the Euclidean measure on [a , b). I n that case , LP(T, fi) i s denoted L p(a, b).
As another notatio n w e mention L P(R).
Exercises.1 . Verif y (usin g th e inequalit y a + b < 2max(a , b) fo r a — xi,
6 = yi) tha tZ p (similarly, L v(T,n)) i s a vector space .
2. Verif y tha t th e convergenc e f n— » / i n C(a , 6) i s th e unifor m convergenc e
/«(*) = 4 f{t).
3. Verif y tha t C(a, b) i s everywhere dens e in L p(a,b) ( 1 < p < oo).
4.7. Th e algebr a B(X) . Le t X b e a norme d space . Fo r eac h a G L(X) =
End X w e set
(4.27) ||a| | = su p-^—r^- = su p ||ax|| .
x^O X || x|| = i
An operato r a G EndX i s said t o b e bounded i f ||a| | < oo , i.e. , ||ax| | < Cx fo r
each x G X (wher e C > 0 is a constant) .
It i s eas y t o chec k tha t th e boundednes s o f a G L(X) i s equivalen t t o th e
continuity o f a o n X . Th e spac e B(X) o f al l bounde d operator s a G L(X) i s a
normed spac e wit h respec t t o nor m (4.27) . Furthermore , B(X) i s an algebr a ove r
F suc h tha t
(4.28) || a&|| < ||a| | • ||b||
for al l a,b G B(X). Repeatedl y applyin g (4.28 ) w e have
(4.29) ||a n
|| < a n
for eac h n G Z+ . Not e tha t ||1 | | = 1 .
If X i s complete, the n B(X) i s also complete (see , for example , [43] , [66]) . I n
that case , i t i s easy t o check , usin g (4.29) , that fo r eac h serie s (4.1 ) (wit h n = oo )
operator serie s (4.2 ) converge s i n B(X) whe n ||a| | < r(/) , wher e r(f) i s the radiu s
of convergence o f (4.1 ) .
Let a G B(X) an d le t $ a b e th e algebr a o f powe r serie s (4.1 ) wit h radiu s o f
convergence r(f) > a. I t i s eas y t o chec k tha t th e ma p 7 : f(x) -* f(a) i s a
functional calculu s of class $ a .
Examples.1 . Th e function s /(A ) = ( 1 — A)- 1
expand s int o a geometri c
progression wit h radiu s o f convergence 1 . Therefore , th e operato r
00
(4.30) /(a ) = £ V
n=0
is define d wheneve r ||a| | < 1 . A direc t calculatio n show s (exercise ) tha t f(a) —
( l - a ) " 1
.
2. Th e exponentia l /(A ) = e x
i s given b y a power serie s with radiu s o f conver -
gence oo . Therefore , th e operato r exponentia l
00 n
(4.31) f(a) -e^ = J2~
n=0
is defined fo r eac h a G B(X).](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-42-320.jpg)
![26 1. BASI C NOTION S
4.8. Th e cas e dim X < oo . Seminorm s p and q on a vector spac e X ar e sai d
to b e equivalent i f
(4.32) p(x) < Aq(x), q(x) < Bp(x)
for each x Gl (fo r som e constants A,B>0). I t i s easy to check (see , for example ,
[72]) that , whe n dim X < oo , all norms o n X ar e equivalent .
Using inequalities (4.32 ) for the norms p and q on X w e find that p-convergenc e
Jbfi ' Jb l o equivalen t t o ^-convergenc e x n— » x. Therefore , t o verif y convergenc e
in X (dim X < oo ) i t suffice s t o fix one of the norm s (4.21 ) o r (4.25) .
Therefore, B(X ) = L(X ) fo r eac h nor m o n X (i.e. , al l linea r operator s ar e
continuous whe n dim X < oo) . Consequently , w e can conside r functiona l calculu s
of class <£ a (se e 4.7) whe n a G L(X).
For example , exponentia l (4.31 ) i s defined fo r eac h a G L(X).
Exercises. 1 . Suppos e dim X < oo . Verif y tha t definitio n (4.2 ) wit h / G &a
(n = oo) coincide s wit h definitio n (4.7) .
2. Verif y (usin g (3.4) ) tha t
(4.33) det(e a
) = e t r a
,
where tra (th e trace of a) i s defined (simila r t o 3.1 ) a s trae fo r eac h basis e of X.
3. Verif y (usin g the Jorda n norma l for m o f a) tha t
(4.34) e a
= lim( l + - ) n
.
Henceforth we shall also use the notation B(X, Y) fo r the vector space of all bounded
(i.e., continuous) operator s a G Horn (X, Y).
5. Unitar y space s
5.1. Definition . Le t X b e a vecto r spac e ove r a field F = R,C . A functio n
/ : X x X —> F is called a Hermitian form o n X i f for all a, (3 G F and all x,y,z G X,
(a) f(x,y) = f(y,x), an d
(/?) /(a x + 02/, z) = a/(:r , *) + /?/(y , z).
Here th e overba r denote s comple x conjugatio n i n F. Condition s (a) an d (/3 )
imply tha t / i s antilinear i n the secon d argument , i.e. ,
/(x, ay + pz) = a/(x , y) + fif(x, z).
A Hermitian for m / i s called a scalar product o n X i f it i s positive definite , i.e. ,
(7) /(# , #) > 0 and f(x, x) — 0 implies x = 0 .
The space X i s said to be unitary i f a scalar product / i s defined o n it. I n tha t
case, on e ofte n use s th e simplifie d notatio n f(x,y) — (x,y). Conditio n (7 ) allow s
us to defin e a nonnegative functio n
(5.1) ||x| | = V ^ ) -
If F — M, then th e comple x conjugatio n i n (a ) an d (/? ) i s trivial , an d / i s a
symmetric bilinea r for m o n X . A unitar y spac e ove r R i s als o calle d a Euclidean
space.
5.2. Proposition . Function (5.1 ) is a norm on X [so that X is a normed
space over F). Furthermore,](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-43-320.jpg)
![5. UNITAR Y SPACE S 29
such that yi = xi (i = 1 ,... , n). Th e vector z — x — y is orthogonal t o the syste m
e [z % = 0 fo r eac h i — 1,... ,n). Therefore , x = y + z, wher e y _ L z. Clearly ,
||x||2
= ||y|| 2
+ ||^|| 2
(th e Pythagorean Theorem). Als o y 2
coincide s wit h th e
left-hand sid e of (5.12). Whenc e (5.1 2 ) (a s y 2
< ||x|| 2
).
Applying (5.1 2) , we have that fo r each x G X at most countably many coordi-
nates Xi (i G I) are different from zero.
An orthonormal system e in X i s said to be complete (respectively , closed) if its
linear spa n i s everywhere dens e i n X (respectively , i f Zi = 0 for each i G / implie s
that z — 0). Th e system e is called a n (orthonormal) basis of X i f each x Gl ca n
be written a s a series
(5.13) x = y^Cjej ,
where at mos t countabl y man y coefficient s Q G F are different fro m zero .
Using the continuity o f the scalar produc t o n X w e have that Q = Xi for eac h
i £ I (i.e. , the coefficients i n (5.13) are uniquely determined) . Similarly , one checks
that
(5.14) H 2
= 5>,|2
.
iei
Examples. 1 . Lete^ G h(^) b e the vector wit h coordinate s 5ij (j G I). The n
the syste m e — (e^)^/ i s an orthonormal basi s in hi^u).
2. Th e function s
(5.15) e n{t) = -^=e M
,
V Z7 T
where n G Z, form a n orthonormal syste m i n 1/2(0, 27r).
We shal l no w show tha t (5.1 5 ) i s a complete system . Indeed , th e linea r spa n
T(0,27r) o f (5.1 5 ) i s th e algebr a o f trigonometri c polynomial s i n L 2(0,27r). Ac -
cording to the Weierstrass Theorem , T(0 , 2n) i s everywhere dens e in (7(0, 2TT) with
respect t o th e unifor m metric . Bu t the n i t i s also everywher e dens e wit h respec t
to th e metri c i n 1 ^(0 , 27r). O n th e othe r hand , C(0,27r ) i s everywher e dens e i n
1/2(0, 2TT) (se e 4.6). W e now conclude that T(0, 2TT) is everywhere dense in 1/2(0, 2ir).
5.5. Hilber t spaces . A unitary spac e H i s called a Hilbert space if it is com-
plete (wit h respec t t o norm (5.1 )) .
In particular , al l unitary space s considere d i n 5.1 and 5.2 are Hilbert spaces .
Unitary space s X, Y ove r a field F ar e said t o be isomorphic i f there exist s a
linear isomorphis m u: X — • Y preservin g th e scalar product , i.e. ,
(5.16) [ux.uy) = (x,y )
for al l x,y G X.
Exercises.1 . Conditio n (5.1 6 ) i s equivalent t o u bein g a n isometry, i.e. , t o
the conditio n ux = ||x| | for each x G X. [Hint : Us e the polarization rule. ]
2. I f a unitary spac e X i s isomorphic to a Hilbert spac e Y, then X i s a Hilbert
space.
5.6. Lemma . Let H be a Hilbert space, e — (e^)^/ an orthonormal system
in H, and Xi (i G I) a system of numbers such that series (5.1 2 ) converges. Then](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-46-320.jpg)
![30 1. BASIC NOTION S
there exists a vector
(5.17) x = J2 x%Ci
iei
in H such that x % (i G /) are the coordinates of x (with respect to e). In that case,
Bessel inequality (5.1 2 ) becomes equality (5.1 4) .
PROOF. Recal l (se e 5.3 ) tha t a t mos t countabl y man y number s X{ (i G /) ar e
nonzero. Therefore , i t suffice s t o conside r countabl e serie s of the for m
oo
(5.18) x = 2_. x
%e
i-
2 = 1
Let s n (n G N) b e a partial su m o f series (5.1 8) . B y the Pythagorea n rule , we have
that s n — 5m||2
equals the corresponding remainder o f series (5.1 2) . I t no w follow s
(as H i s complete) tha t serie s (5.1 8 ) converges .
The othe r assertion s o f this lemm a were proved i n 5.4 . •
Now we can stat e an d prov e th e mai n theore m o f Hilbert spac e theory .
5.7. Theorem . Each Hilbert space H has an orthonormal basis. Moreover, if
e is an orthonormal system in H, then the following are equivalent:
(i) e is complete;
(ii) e is closed]
(iii) e is a basis of H.
PROOF. Th e orthogonalit y relatio n z A- e (i.e. , Zi — 0 fo r eac h i G I) i s
equivalent t o z 1 H{e), wher e H{e) i s th e closur e o f th e linea r spa n o f e. I f (i )
holds, the n z _ L H. I n particular, z _ L z, i.e. , z — 0 and (i )= > (ii). Th e implicatio n
(ii)= > (iii) follows from Lemm a 5.6 . Indeed , fo r each x G H le t y G H b e the vecto r
defined b y (5.1 7) , wher e X{ are th e Fourie r coefficient s o f x. Then , fo r z — x— y,
we have tha t z _ L e, i.e. , z = 0 . Thu s x = y i n (5.1 7) . Th e implicatio n (iii )= > (i)
follows directl y fro m th e equalit y x — limn sn , wher e th e s n ar e th e partia l sum s
of (5.1 8) . Thu s assertion s (i) , (ii) , and (iii ) ar e equivalent .
Using Zorn' s lemma, we see that ther e is a maximal (wit h respect t o inclusion )
orthonormal syste m e m H. I f 0 ^ z ± e , the n th e uni t vecto r z 0 — ^/IMI extend s
the syste m e , which i s impossible. Therefore , e is a basis of H. •
Example. Syste m (5.1 5 ) i s an orthonormal basi s in L2(0, 2TT). I n other words ,
each functio n / G I/2(0, 2TT) can b e written a s a Fourier serie s
oo
(5.19) /(*) = Yl c
nemt
,
which converge s i n the metri c o f L2(0, 27r), with coefficient s
(5.20) c n = ~J f(t)e- tnt
dt.
Moreover, equalit y (5.1 4 ) become s th e Plancherel formula
(5.21) _ / f(t) 2
dt= ] T c n
•'O <n.= — nn
5.8. Corollary . Each Hilbert space is isomorphic to /2(^) {for a suitable u).](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-47-320.jpg)
![32 . BASI C NOTION S
Example. Le t Ho C H b e a close d subspac e an d Hi = H$ it s orthogona l
complement (i.e. , the se t o f all y G H suc h tha t y J_ HQ). The n
(5.26) H = H 0@H
1 .
A projectio n operato r p o n H i s calle d a n orthoprojector i f (px,y) = (x,py)
for al l #,2 / G iif . Fo r suc h a projector , (px,p'y) = 0 fo r al l x,y G H, wher e
pi — 1 — p (th e complementar y orthoprojector) . Consequently , w e have orthogona l
decomposition (5.26) , where HQ = pH an d Hi — p'H.
A famil y o f orthoprojector s (pi) %^i o n H i s sai d t o b e orthogonal i f p^j = 0
when i ^ j . I t i s easy t o chec k that , i n that case , the forma l serie s
i
converges a t eac h x G H. Moreover , p project s H ont o th e direc t orthogona l su m
of the Hi—piH. I n particular , p — 1(the identit y operator ) i n (5.24) .
Exercises.1 . I f p ^ 0 i s a n orthoprojector , the n ||p| | = 1 (i n particular ,
peB(if)).
2. Eac h nondecreasin g sequenc e o f orthoprojector s p n G B(H) converge s a t
x G H t o a n orthoprojecto r p G B(iJ) . [Hint : Th e famil y Ap n = p n — pn_i
(po = 0 ) i s orthogonal. ]
5.10. Theorem . Any continuous linear functional f G H' is of the form
f(x) = (x,y),
where y e H. Moreover, f — ||y||. The map y ^ f is an antilinear isomorphism
H ^ H f
.
PROOF. Le t H 0 = ker/ . The n codimi7 0 = 1 if / ^ 0 . Thi s gives rise to (5.26 )
with one-dimensiona l Hi = FCQ, wher e 0 ^ e oJ_ #o- Settin g x — xo + Aeo , where
xo G HQ and A G F, w e have
f(x) = A/(e0), (x , e0) = A(e 0, e0),
which implie s (5.26 ) wit h y = /(eo)eo an d ||eo| | = 1 . T o find the nor m o f / w e can
use (5.3 ) (replacin g x b y y). Thi s give s ||/| | — y. D
Exercise. Verif y tha t th e antilinea r isomorphis m H « H' abov e i s given b y
(5.27) (x,y) = {x,y),
where x i- » x i s th e comple x conjugatio n i n H (wit h respec t t o a n orthonorma l
basis i n H).
5.11. Theorem . For each a G B(H) (see 4.7 ) t/ier e exist s a unique (adjoint)
operator a * G B(if), defined by
(5.28) (ax,2/ ) = (a:,a*y )
/or a/ / x,y E H. The map a -^ a* is antilinear and satisfies the relations (ab)* —
6*a* and a** = a . Moreover, a = ||a*|| .
PROOF. Not e tha t f(x) = (ax,y) i s a continuou s linea r functiona l o n H, an d
therefore /(# ) = (x ,2;) fo r som e (uniquel y determined ) z E H. I t i s clea r tha t
z i s a linea r functio n o f y. Settin g z = a*y, w e hav e (5.28) . Recal l als o tha t](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-49-320.jpg)
![5. UNITAR Y SPACE S 33
||a|| = su p ||ax||, wher e th e supremu m i s take n ove r th e uni t spher e x = 1 .
Therefore,
||a|| =sup|(ax,2/)| = su p |(x,a*y)| = ||a*|| ,
where the supremum i s taken ove r x = ||y| | = 1 . Th e other propertie s of the ma p
a*-* a* follow easil y fro m (5.28) . •
Exercise. A functio n / : H x H— » F i s said t o b e conditionally bilinear (o r
sesquilinear) i f /(#, y) i s linear in x and antilinear i n y. Prov e that eac h continuou s
conditionally bilinea r functio n / : H x H — > F i s o f th e for m (5.28 ) (fo r som e
aeB(H)).
5.12. Definition . A n operato r a G B(if) i s said t o b e (respectively ) Hermit-
iarij anti-Hermitian, unitary i f a* = a, a* = —a , a* = a - 1
(i n the latte r case , a is
assumed t o b e invertibl e i n 3(H)).
It is easy to check (exercise) that a * = a if and only if the form f(x,y) = (ax , y)
is Hermitian and also if and only if the quadratic form <p(x) = (ax , x) is real-valued.
Moreover, a * = a i f an d onl y i f matri x (5.22 ) i s Hermitian (fo r eac h orthonorma l
basis o f H).
The substitution ai
—> ia (whe n F — C) yields a bijection between real subspaces
of Hermitia n an d anti-Hermitia n operator s i n B(i7) . Fo r eac h a G B(jFf) , th e
operators
a
i = o (a
+ a
*)> a 2 = —( a - a* )
are Hermitian , an d w e hav e a uniqu e decompositio n a = a + ia,2- Th e operato r
a G B(jff) i s unitary mean s that a is an automorphis m o f the Hilber t spac e H.
Exercises.1 . I f a = a* , the n eac h eigenvalu e a o f A i s rea l an d an y tw o
eigenvectors wit h differen t eigenvalue s a ^ (3 are orthogonal .
2. IfaHo C HQ (where HQ is a subspace of H), the n a*H$ C H$ . In particular,
decomposition (5.25 ) i s invariant i f a — a*.
3. I f H i s a finite-dimensional unitar y spac e ove r C , the n eac h Hermitia n
operator a G L(H) i s diagonalizable. [Hint : Us e Exercises 1 and 2. ]
4. An y Hermitia n operato r a G B(H) i s uniquely determine d b y it s quadrati c
form (f a(x) = (ax,x) . I n othe r words , a = 6 <
^> ipa = ^ . [Hint : Expres s f a(x,y) =
(ax,y) vi a</? a using (5.8). ]
Example. A n operato r a G B(if) i s said t o b e positive (a > 0) i f (ax,x ) > 0
for eac h x G fl". I n particular , a = a* , i.e. , a i s Hermitian . I f d i m # < oc , the n
a — diag (ai,... , an ), wher e c ^ > 0 ( z = 1 ,.. . ,n).
5.13. Compac t operators . A linear operator a G L(i7) is said to be compact
if i t send s ever y bounde d se t A C H t o a precompac t subse t o f H. Equivalently ,
if Si i s the uni t bal l in H (i.e. , the set o f all x G H suc h that ||x| | < 1 ) , then aS i s
precompact.
Here w e us e th e followin g definitio n o f precompactnes s i n a metri c spac e H:
a subse t N C H is precompac t i f it s closur e N i s compact . I n particular , eac h
precompact set N C H i s bounde d (wit h respec t t o th e nor m o n H). W e no w
conclude that an y compac t operato r a G L(iJ) i s bounded, i.e. , a G B(H).
Example. Le t H = Z/2(T, /i), where /x(T ) < oc . A n integra l operato r
(5.29) (af)(t)= [ K(t,s)x(s)dn(s),](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-50-320.jpg)
![34 . BASI C NOTION S
with K G L2(T2
,/i2
), i s called a Fredholm operator o n H.
It i s eas y t o chec k (see , fo r example , [66] ) tha t Fredhol m operato r (5.29 ) i s
compact. Moreover , th e Fubin i theore m implie s tha t a ha s a n adjoin t Fredhol m
operator a* , generate d b y the functio n K*(t,s) = K(s,t). A s a direct consequenc e
of (5.29) , we have a n estimat e
(5.30) ||a| | < HATH,
where ||i^| | i s the norm of K i n L2(T2
, /i2
). Finally , the Fubini theorem als o implies
that, fo r eac h orthonorma l basi s e n (n G N) o f H, th e matri x element s a mn —
(aen, e m) o f a ar e given b y
where e mn(£, s) — ern(t)en(s). Th e parenthese s i n th e right-han d sid e o f (5.31 )
stand fo r th e scala r produc t i n L 2(T2
,/x2
).
Henceforth w e assum e tha t F = C . A basi s e = (ei)i ei o f H i s calle d a n
eigenbasis wit h respec t t o a £ B(H ) i f aei = a ^ fo r al l i G /.
5.14. Theore m (Hilbert) . Any compact Hermitian operator a G B(i7) has an
orthonormal eigenbasis in H. In other words,
(5.32) # = 0 #A
A
(direct orthogonal sum), where H = ker(a — A). Furthermore:
(i) If X ^ 0, then dimH < oc (i.e., the eigenvalue X has finite multiplicity).
(ii) The number of different eigenvalues of a is either finite or countable.
(iii) / / the number of eigenvalues of a is countable, then they form a sequence
such that X n— > 0.
For a proof, se e D.4. Se e also [43 ] an d [66] .
5.15. Corollary . / / K* = K, then the Fredholm operator (5.29 ) has an or-
thonormal eigenbasis en (n G N). Furthermore:
(a) The eigenvalues a n (n G N) are real and
oo
(5.33) J2 a
n<™-
7 1 = 1
(/?) If the integrals J(t) — f K(t,s) 2
dfi(s) are uniformly bounded, then each
function y — ax (y G im a) expands into a (generalized) Fourier series
oo
(5.34) y(t) = ^2 <*nXne n(t),
n=l
which converges absolutely and uniformly.
Indeed, i f en (n G N) i s an eigenbasi s o f (5.29) , then (5.31 ) implie s tha t
oo
(5.35) K(t, s) = ^2 a nen(t)en(s).
7 1 = 1
In particular , th e left-han d sid e of (5.33 ) coincide s with ||i^|| 2
, whenc e (a).](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-51-320.jpg)
![6. TENSO R PRODUCT S 35
To prov e (/?) , notic e that , fo r a fixed t , th e number s a nen(t) ar e th e Fourie r
coefficients o f (5.35) . Applyin g th e Besse l inequality, w e have
oo
]T|an en (£)|2
< J(t)<C.
n=l
Now w e ca n appl y th e Cauchy-Bunyakovsk y inequalit y t o verif y th e convergenc e
of series (5.34) . A s a result, we have (/?) , i.e., series (5.34 ) converge s absolutely an d
uniformly.
Assertion (/? ) i s usually calle d th e Hilbert-Schmidt Theorem.
6. Tenso r product s
6.1. Theorem-Definition . For any pair of vector spaces AT, Y over a field F
there exists a unique, up to isomorphism, vector space X ® Y, called the tenso r
product o f X an d Y, which is defined by the following conditions.
(a) There is a bilinear map n: X x Y —* X 0 Y, (x , y)i
— » x ® y, whose image
generates X 0 Y.
(j3) For any pair of linearly independent systems e , / in (respectively) X , Y the
system e® f consisting of vectors x®y, where x G e, y G /, is linearly independent
in X ®Y.
In addition, as follows from (a) and ((3), X ® Y has the following universal
property in VECT^? :
(7) Any bilinear map a: X xY — » Z (in VECT^?) has a unique extension to
a linear map j3: X ®Y — > Z given by
(6.1) a{x,y) = fi(x®y),
where x G X and y <EY. Finally, X(&Y is uniquely determined (up to isomorphism)
by conditions (a) and (7) .
PROOF. Le t F[X xY] b e th e vecto r spac e over F wit h basi s X xY (se e 2.1).
We define X<S>Fa s th e quotien t spac e o f F[X x Y] b y th e subspac e N generate d
by the element s
(6.2) (Xx + fix, y) - (x, y) - /i(x' , y),
(6.3) (x , Xy + /V ) - H x
i v) - M^ > y'),
where A,/iGF , X,X' G X, an d y,y f
G Y. B y thi s definition , th e canonica l projec -
tion 7r : F[X xY] — > X <S> Y i s bilinear an d cover s X ®Y. Settin g 7r(x , y) = x 0 y,
we have (a). Nex t w e want t o sho w that conditio n (7 ) i s satisfied .
Indeed, (6.1 ) define s [5 on the basi s 1x 7, whic h gives rise to a linear ma p o n
F[X x Y] (se e 2.3) . Sinc e a i s bilinear, w e have tha t (3 annihilates element s (6.2 )
and (6.3) , i.e. , /3(N) = 0 . The n (3 i s defined o n th e coset s (modAQ , i.e. , i t ca n b e
viewed a s a linear operato r X 0 Y — » Z. Thi s implie s (7) .
In particular , fo r eac h pai r o f linea r operator s a G Hor n (X, X') an d b G
Hom(F, Y') (i n VECTi? ) th e ma p a(x,y) = ax ® by i s bilinear , an d therefor e
gives rise to a linea r operato r (3 = a 0 6 G Horn (X 0 Y , X' 0 Y') suc h tha t
(6.4) (a (8) b)(x ® y) = ax ® by,
where x G X an d y G Y. Late r i n th e book , w e shal l discus s th e connectio n
between th e symbo l a 0 b and th e tenso r produc t symbo l i n (a). Similarly , fo r an y](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-52-320.jpg)
![—Trans.
II.1 Accommodation is here so difficult to procure, that
the senators are obliged to sleep three upon one
mattress laid upon the floor: their food consists, it is
said, almost without exception of salted pork.
II.2 In this part of the country, they have either
feather beds or moss mattresses; if these latter are
old, the moss clots together, and it is like lying on
cannon-balls.
II.3 [This is the same corps which the Philadelphians
extol so highly, that one might almost suppose them
equal to the artists of the Theatre Français, if,
unfortunately, one visit to the theatre, did not
completely dispel the illusion!]
II.4 Among the slave traders, a Hollander from
Amsterdam, disgusted me particularly, his name was
Jacobs. He had the most vulgar and sinister
countenance imaginable, was constantly drunk, and
treated the wretched negroes in the most brutal
manner; he was, however, severely beaten by these
miserable beings, driven to despair. II.4a
II.4a The virtuous indignation of the Duke, at these horrible
consequences of slavery, is such as every man, not hardened
by long familiarity with such scenes, must feel; those to
whom they are daily presented regard them with calm
indifference, or even attempt to argue in favour of their
continuance and harmlessness. It is not as generally known,
as it should be, that the slave trade is carried on, almost as
vigorously now, as ever it was, and by citizens of almost
every nation; not in the least excepting Americans. The slave
vessels sail principally from Havanna and St. Thomas, and
land their cargoes on the island of Puerto Rico, and
elsewhere, whither purchasers and agents resort, when such](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-61-320.jpg)
![—Trans.
an arrival occurs. Two schooners, with large cargoes, arrived
in Puerto Rico in February last, and two brigs were daily
expected. It is said in the West Indies, that all ships of war, of
powers owning West India Colonies, connive at the trade,
which is fully supported by facts; as French, Danish, and
English cruisers were in the vicinity, when the above
mentioned cargoes arrived. The idea of cruising off the coast
of Africa, to prevent the trade, is ridiculed by the slave
dealers, with one of whom the writer of this note conversed.
If the American, or any other government really wished to
put an end to this trade, it could be very effectually
accomplished, by sending small armed vessels to intercept
the slave traders near their places of landing cargoes, which
are not very numerous. It is also said, in the West Indies,
that the Havanna traders still contrive to introduce Africans
into the southern part of the United States; of the truth or
falsehood of this, we know nothing. The slave vessels are
generally Baltimore clipper brigs, and schooners, completely
armed and very fast sailers. Two of them sailed on this
execrable trade in February last, from a part visited by the
writer.—Trans.
II.5 Colonel Croghan was one of the most distinguished
officers of the American army. In the last war, he
defended a miserable stockade, (Fort Stephenson) on
Lake Erie, against a force eight times greater than his
own, which had artillery, and drove it back. After the
peace, he was appointed Post Master of New Orleans,
and during my stay in this city, he had entered again
into the army, and held the post of second Inspector-
General.
II.6 [Commonly known in New Orleans by the name of
the Calaboose, (from Calabozo, the Spanish term for a
vaulted dungeon,) and a great terror to evil-doers in
that city; the efficiency of the police of which can
never be enough admired.]
II.7 I have already made some remarks with regard to
the apprehension of negroes in Charleston. If a person
wishes here to have a house-negro, male or female,](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-62-320.jpg)
![—Trans.
chastised, they are sent with a note, in which the
number of lashes which the bearer is to receive are
specified, with a quarter of a dollar; he or she is
lodged in the slave prison. Here the slave receives the
punishment, and a certificate, which he must carry to
his master. The maximum of lashes is thirty-nine,
according to the Mosaic law. The species of
punishment is specified as in Charleston, or “aux
quatre piquets.” In this last case, the poor wretch is
pressed out flat on his face upon the earth, and his
hands and feet bound to four posts. In this posture he
receives his flogging. This frightful method of
chastisement, is principally in use on the plantations;
and cruel discipline is there chiefly practised. Whoever
wishes to punish a house servant severely, either hires
or sells him to work on the plantations.
II.8 If it be known that a stranger, who has
pretensions to mix with good society, frequents such
balls as these, he may rely upon a cold reception from
the white ladies.
II.9 [A plain, unvarnished history of the internal slave
trade carried on in this country, would shock and
disgust the reader to a degree that would almost
render him ashamed to acknowledge himself a
member of the same community. In unmanly and
degrading barbarity, wanton cruelty, and horrible
indifference to every human emotion, facts could be
produced worthy of association with whatever is
recorded of the slave trade in any other form. One of
these internal slave traders has built, in a neighbouring
city, a range of private prisons, fronting the main road
to Washington, in which he collects his cattle previous
to sending off a caravan to the south. The voice of
lamentation is seldom stilled within these accursed
walls.]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-63-320.jpg)
![—Trans.
—Trans.
—Trans.
II.10 This Frenchman, a merchant’s clerk from
Montpelier, was not satisfied with this: he went to the
police, lodged a complaint against the girl, had her
arrested by two constables, and whipped again by
them in his presence. I regret that I did not take a
note of this miscreant’s name, in order that I might
give his disgraceful conduct its merited publicity.
II.11 [Nonsense.]
II.12 [Our author has somehow been confused in his
diary here: the mouth of La Fourche is generally called
seventy-five miles above New Orleans, Stoddart makes
it eighty-one. At any rate it is about half way between
Bayou Sara or Point Coupee and the city of New
Orleans; and of course the Duke must have passed
Donaldsonville, which is at the junction of La Fourche
with the Mississippi, in the morning of the day he
passed Baton Rouge.]
II.13 In these rivers there is a difference understood
between the two kinds of trunks of trees which lie in
the stream, and are dangerous to vessels, i. e. snags
and sawyers. The first, of which I have spoken already
in the Alabama river, are fast at one end in the bottom,
and stand up like piles; the others are not fastened, by
being moved by the current the upper end of the tree
takes a sawing motion, from whence its appellation is
derived.
II.14 Coluber coccineus.
II.15 [These log turnpikes are better known by the
name of “corduroy roads.”]
II.16 [This is, perhaps, the most charitable idea that
can be formed of the actions of such reformers, as well
as of a “lady” heretofore mentioned, who has unsexed
herself, and become so intoxicated with vanity, as](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-64-320.jpg)
![—Trans.
—Trans.
—Trans.
—Trans.
—Trans.
—Trans.
—Trans.
—Trans.
enthusiastically to preach up a “reformation” in favour
of the promiscuous intercourse of sexes and colours,
the downfall of all religion, and the removal of all
restraints imposed by virtue and morality!]
II.17 [It is understood that Mr. M‘Clure has long since
given up all connexion with the New Harmony bubble.]
II.18 By late newspapers it appears, that the society
actually dissolved itself, in the beginning of the year
1827.
II.19 [According to the report of some females, who
were induced to visit New Harmony, and remained
there for some time, any situation much above abject
wretchedness, was preferable to this vaunted
terrestrial paradise.]
II.20 He was drowned in the Wabash, which he
attempted to swim over on horseback.
II.21 [He is at this time advertising a boarding-school
in the Western country, on his own account, which is
to be under his immediate superintendence!]
II.22 [He has left it some time since, as well as Dr.
Troost.]
II.23 These had been presented to Bishop Fenwick by
Cardinal Fesch, for his cathedral, and were only here,
until they could find their place in the Temple of God.
II.24 [Brother of Lord Hill.]
II.25 [Peyton Symmes, Esq. receiver of the land
office.]
II.26 [Tetrao Umbellus, L.]
II.27 These meadows are designated in America, by
the name of prairies, and extend over large tracts of](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-65-320.jpg)
![—Trans.
—Trans.
land in the western country; they are covered with
high grass; trees grow very sparingly on them, while
the surrounding forests exhibit the most beautiful
trees; the soil of these prairies generally consists of
turf-moor.
II.28 [The Ohio Eagle.]
II.29 Knopendraayerye.
II.30 In the year 1826, I enjoyed three springs; the
first about the end of February at New Orleans, the
second at New Harmony and Louisville, and the third
in the state of Ohio, and west Pennsylvania.
II.31 In this hymn-book are some pieces, which, if the
perfect child-like innocence of these maidens be not
recollected, might appear rather scandalous. For
instance, there is a literal translation of the song of
Solomon, among others.
II.32 [It is to be hoped that the able and luminous
report of the commissioners appointed by the state, to
make inquiries on the subject of penitentiary discipline,
will be sufficient to correct the glaring errors of this
new system; which like most of the new systems of
the present day, is clearly proved thereby to be more
specious than beneficial. The evidence accumulated by
the commissioners is of a character to satisfy every
candid mind, not chained to the support of a particular
theory, that solitary confinement without labour, is
unequal in operation, inadequate to the end proposed,
and promises to be as destructive to human life as it is
discordant to humane feelings.]
II.33 [This is a very erroneous idea. The taste for
painting and music has not been cultivated, generally,
in this country, on account of the condition of property
and society, which demand of Americans a primary](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-66-320.jpg)
![—Trans.
—Trans.
—Trans.
—Trans.
devotion to things absolutely necessary and useful. As
wealth becomes more accumulated, artists will be
encouraged; and then we have no fear of their being
long inferior to the artists of any other nation.]
II.34 [The reason is, that the portrait painter ministers
to the gratification of personal vanity, or self-love, and
the landscape painter to a refined taste. As the
proportion of egotists to men of refined and cultivated
taste, is somewhat less than a million to one, it is easy
to see which branch of the arts will receive most
attention.]
II.35 Kensington was formerly a distinct village, on the
Delaware above Philadelphia; the city has now
extended thus far, so that it now belongs to the city.
The tree was some years ago struck by lightning and
destroyed.
II.36 In service of the Netherlands, on his return from
a mission to the new South American republics.
II.37 Austrian Consul-General to the United States.
II.38 Whose acquaintance I made on the Mississippi
during my trip from Louisville to Cincinnati.
II.39 [This light is emitted by molluscous animals,
which are exceedingly abundant in some parts of the
ocean. They are also seen to great advantage during
the night, in the Chesapeake bay.]
II.40 A respectable London merchant, and native of
Flanders, to whom I am much indebted for very
important services.
II.41 [Now exhibiting in New York.]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-67-320.jpg)
![Spelling and Typography
Spelling was corrected if the mistake was clearly mechanical, or
inconsistent with the author’s (or translator’s) usage elsewhere. This
includes some spellings that were acceptable in 1828, but are different
from other occurrences of the same word.
All commas are as printed. Inconsistent italicization of ships’ names is as
in the original. The notation “invisible” means that there is an
appropriately sized empty space, but the punctuation itself is missing.
Some specifics:
“Bodleïan” is written with dieresis
“chesnut” is standard for the time and is used consistently
“team boat” (referring to canals) is not an error
“lime-stone” and “sand-stone” are hyphenated at the beginning of
the book but later become single words
“free-stone” or “freestone” are not frequent enough to establish a
pattern
“country seat” starts out as two words, but later becomes
hyphenated “country-seat”
the inconsistent spacing of “no( )where” “every( )where” and “else(
)where” is unchanged
“back ground” is generally two words; “fore-ground” (with hyphen)
occurs only once
Spelling (unchanged):
appointed him his aid
present as aid to the Emperor
Mr. Butler, his aid
[the spelling aid is used consistently]
the tatoed and dried head of a New Zealand chief
roast-beef, plumb-pudding, &c.
This place is called the antichamber.
it is fixed in a cramp [text unchanged: error for clamp?]
some negroes, who were frolicing during the Christmas holy-days
corset inventress to the Dutchess of Kent [Duchess and
Dutchess are each used once]
French (corrected):](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-69-320.jpg)
![From the left wing a line runs en crémaillère [crémaillére]
When a lady is left sitting, she is said to be “bredouille.”
[bredouillè]
so that they would consider their labour in the light of a corvée.
[corveè]
Errors (corrected):
strange bas-reliefs, representing ancient hunting scenes [bass-
reliefs]
the rooms are not large; the beds [is not]
On the top of the capitol is a cupola [cupalo]
the pen is too feeble to delineate the simultaneous feelings of
insignificance and grandeur [simultaneons ... insignificence]
then a company of sappers and miners, [minors]
in which again each sex has its own side [each sect]
There are two paintings by Teniers [painting]
the office and place of deposit for bound bibles [deposite]
They recall to memory Glenn’s Falls on the Hudson [recal; recall is
used consistently elsewhere]
the other loses by faint colouring [looses]
an English copy of the illustrations of Göthe’s Faust laid open.
[illustratrations; spelling Göthe with umlaut unchanged]
In a hollow place there is a basin, or rather a reservoir [their is]
close by it stands the prison, or county gaol, [goal]
this journey of one hundred and ninety-eight miles [ninty-eight]
the log houses were only employed as negro cabins [onegr]
we might lie several days, perhaps weeks here [several day]
because they had not received their pay for some time. [missing
not]
Several of the French families here settled [familes]
so as not to lose themselves in the woods [loose]
and on this account, the proceeding to me appeared arbitrary
[acount]
has followed me even in America like an evil genius [and evil]
an inspection on the Red river, the Arkansas, and New Orleans
[Arkansa; Arkansas is used consistently elsewhere]
a great part of the houses are built of brick; [missing a]
We frequently rode along the new national turnpike road
[frequently road]
over some stone bridges of sumptuous construction [contruction]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-70-320.jpg)
![about the right bank of the Alleghany and Ohio [righ]
parallel ridges, called Laurel hill. [callel]
the girls learn to sew and knit [sow]
The gentlemen above named accompanied me to the vessel.
[accompaned]
every thing, manufactured in Birmingham, [Burmingham]
a fine view into a considerable suite of rooms [considerble]
until no poison is left in the stomach [stomuch]
Spacing, hyphenization, capitalization:
By his highness, Bernhard, Duke of Saxe-Weimar Eisenach.
[capitalized as shown]
two stories besides a ground floor, and may contain [floor,and]
so that it is excellently adapted to waterworks [anomalous missing
hyphen unchanged]
They have a large kitchen garden [they have]
The corpse is put in the corpse-house [the corpse]
views of Monticello, Mount Vernon, the principal buildings in
Washington [Mount-Vernon]
many evergreen trees and bushes. [ever green]
with Madam Herries; he is a Frenchman [Herries;he]
This is, however, the case with most of the stores [this is]
Punctuation:
Footnote I-4: [This manuscript .... of London.]—Trans.
[printed of London.—Trans.]: changed for consistency]
On the 9th, at 6 A. M. she arrived at Falmouth. [6 A. M]
Schenectady.— Utica.— Rochester. [—Utica—]
and produce much vexation in consequence of the baggage. [final
. missing]
the village of St. Regis, the last belonging to the United States.
[final . missing]
a monument erected by the colony in honour of Lord Nelson. [final
. missing]
to protect the place of embarkation by a fort. [final . missing]
On the ensuing morning I went with Mr. Halbach to Mr. Vaux [Mr
Vaux: period invisible]
mineralogy and geology.— ... lectures on chemistry. [missing .
after geology and chemistry]
drawing of the human figure.— [final . missing]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-71-320.jpg)
![The cotton cleaned from its seed is put into a large chest, pressed
in, and packed up. [final . missing]
Mr. Nott studied in England and France [Mr Nott: period invisible]
the 16th ultimo from Liverpool [ultimo.]
[Footnote II-9: [... within these accursed walls.]—Trans. [missing
— before Trans.]
the river is fordable in many places above the falls.” [close quote
missing]
a diameter of one hundred and fifty yards during forty miles.”
[close quote missing]
even this navigation so expensive and destructive to the wood, will
cease [punctuation unchanged]
one hundred and twenty horses, which daily work here, [work
here.]
in the neighbourhood of Lake Ontario and the river St. Lawrence
[St Lawrence]](https://image.slidesharecdn.com/16673308-250530142014-5549899a/85/Principal-Structures-And-Methods-Of-Representation-Theory-Zhelobenko-72-320.jpg)
Principal Structures And Methods Of Representation Theory Zhelobenko
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- 8. Translations o f MATHEMATICAL MONOGRAPHS Volume 22 8 Principal Structure s and Method s of Representation Theor y D. Zhelobenk o Translated b y Alex Martsinkovsk y American Mathematica l Societ y !? Providence , Rhod e Islan d °^VDED^* 10.1090/mmono/228
- 9. E D I T O R I A L C O M M I T T E E A M S S u b c o m m i t t e e Robert D . MacPherso n Grigori i A . Marguli s Jame s D . Stashef f (Chair ) A S L S u b c o m m i t t e e Steffe n Lemp p (Chair ) I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair ) H . I I . >Kejio6eHK O O C H O B H b l E C T P Y K T Y P b l H M E T O H b l T E O P M M r i P E H C T A B J I E H M M MIIHMO, MOCKBA , 200 4 This work wa s originally publishe d i n Russian b y MIIHM O unde r th e title "OcHOBHbi e CTpyKTypu TeopH H npe,a;cTaBJieHHM " ©2004 . Th e presen t translatio n wa s create d unde r license fo r th e America n Mathematica l Societ y an d i s published b y permission . Translated fro m th e Russia n b y Ale x Martsinkovsk y 2000 Mathematics Subject Classification. Primar y 20-01 , 20Cxx ; Secondary 1 7B1 0 , 20G05 , 20G42 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mmono-228 Library o f Congres s Cataloging-in-Publicatio n Dat a Zhelobenko, D . P. (Dmitri i Petrovich ) [Osnovnye struktur y i metody teori i predstavlenii . English ] Principal structure s an d method s o f representatio n theor y / D . Zhelobenk o ; translate d b y Alex Martsinkovsky . p. cm . — (Translation s o f mathematical monograph s ; v. 228) "Originally publishe d i n Russian b y MTSNMO unde r th e title 'Osnovny e struktur y i metod y teorii predstavlenii ' c2004"—T.p . verso . Includes bibliographica l reference s an d index. ISBN 0-821 8-3731 - 1 (alk . paper) 1. Representation s o f groups. 2 . Representations o f algebras. I . Title. II . Series. QA176.Z5413 200 4 512/ .22—dc22 200505235 2 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapter fo r use in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given. Republication, systemati c copying , or multiple reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e mad e b y e-mail t o reprint-permission@ams. org. © 200 6 b y the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d t o the United State s Government . Printed i n the United State s o f America . @ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guidelines established t o ensure permanenc e an d durability . Visit th e AMS home pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 1 10 09 08 07 06
- 10. Contents Preface i x Part 1 . Introductio n Chapter 1 . Basi c Notion s 3 1. Algebrai c structure s 3 2. Vecto r space s 8 3. Element s o f linear algebr a 1 4 4. Functiona l calculu s 2 0 5. Unitar y space s 2 6 6. Tenso r product s 3 5 7. 5-module s 4 0 Comments t o Chapte r 1 4 6 Part 2 . Genera l Theor y 4 9 Chapter 2 . Associativ e Algebra s 5 1 8. Algebra s an d module s 5 1 9. Semisimpl e module s 5 8 10. Grou p algebra s 6 4 11. System s o f generators 7 0 12. Tenso r algebra s 7 5 13. Forma l serie s 8 0 14. Wey l algebras 8 6 15. Element s o f ring theory 9 3 Comments t o Chapte r 2 9 8 Chapter 3 . Li e Algebras 9 9 16. Genera l question s 9 9 17. Solvabl e Li e algebras 0 5 18. Bilinea r form s 0 9 19. Th e algebra. /7(g) 5 20. Semisimpl e Li e algebras 1 2 0 21. Fre e Lie algebras 2 5 22. Example s o f Lie algebras 1 3 0 Comments t o Chapte r 3 3 7 Chapter 4 . Topologica l Group s 1 3 9 23. Topologica l group s 3 9 24. Topologica l vecto r space s 1 4 5
- 11. CONTENTS 25. Topologica l module s 26. Invarian t measure s 27. Grou p algebra s 28. Compac t group s 29. Solvabl e group s 30. Algebrai c group s Comments t o Chapte r 4 lapter 5 . Li e Group s 31. Manifold s 32. Li e group s 33. Forma l group s 34. Loca l Li e group s 35. Connecte d Li e group s 36. Representation s o f Lie group s 37. Example s an d exercise s Comments t o Chapte r 5 152 157 164 170 175 181 185 187 187 192 198 203 209 214 219 224 Part 3 . Specia l Topic s Chapter 6 . Semisimpl e Li e Algebra s 38. Carta n subalgebra s 39. Classificatio n 40. Verm a module s 41. Finite-dimensiona l g-module s 42. Th e algebr a Z(g) 43. Th e algebr a F ext(g) Comments t o Chapte r 6 Chapter 7 . Semisimpl e Li e Group s 44. Reductiv e Li e group s 45. Compac t Li e group s 46. Maxima l tor i 47. Semisimpl e Li e group s 48. Th e algebr a A(G) 49. Th e classica l group s 50. Reductio n problem s Comments t o Chapte r 7 Chapter 8 . Banac h Algebra s 51. Banac h algebra s 52. Th e commutativ e cas e 53. Spectra l theor y 54. C*-algebra s 55. Representation s o f C*-algebra s 56. Vo n Neumann algebra s 57. Th e algebr a C*(G) 58. Abelia n group s Comments t o Chapte r 8 225 227 227 233 238 244 250 256 262 263 263 268 272 277 283 289 294 300 301 301 307 312 317 323 329 335 340 346
- 12. CONTENTS Chapter 9 . Quantu m Group s 59. Hop f algebra s 60. Wey l algebra s 61. Th e algebr a U q($) 62. Th e categor y 0 nt 63. Th e algebr a A q($) 64. Gaussia n algebra s 65. Projectiv e limit s Comments t o Chapte r 9 Appendix A . Roo t System s Comments t o Appendi x A Appendix B . Banac h Space s Appendix C . Conve x Set s Appendix D . Th e Algebr a B(H) Bibliography Index vii 347 347 353 359 365 370 377 383 388 391 402 403 407 413 421 425
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- 14. Preface The titl e o f this boo k admit s tw o interpretations, wit h emphasi s o n either th e "principal structures" o r the "representatio n theory" . Th e latter is more preferable , as i t i s difficult t o identif y wha t th e basi c structure s o f moder n mathematic s are . Nevertheless, i n a sense, the tw o interpretations agree . Indeed, representatio n theor y deal s wit h fundamenta l aspect s o f mathemat - ics, beginnin g wit h algebrai c structure s lik e semigroups , groups , rings , associativ e algebras, Lie algebras, etc. Eventuall y topolog y enter s th e pla y b y way of algebro - topological and algebro-analytica l structure s lik e topological groups, manifolds, Li e groups, etc. Formall y speaking , the subject o f representation theor y i s the study of homomorphisms (representations ) o f abstract structure s int o linear structure s con - sisting, a s a rule , o f linea r operator s o n vecto r spaces . Bu t i n fac t representatio n theory i s tie d u p wit h structur e theory . Ver y earl y th e student s o f mathematic s learn tha t "rin g theor y i s inseparably linke d wit h modul e theory" . A n importan t feature o f this settin g i s that th e abov e structures ar e either linea r o f have suitabl e linearizations (linea r hull s of semigroups, tangen t Li e algebras of Lie groups, etc.) . Here we come to the question of the role representation theor y plays in moder n mathematics. Originall y (i n the beginning of the 20th century) tha t rol e was rathe r modest an d was confined t o the representation theor y o f finite groups and , eventu - ally, finite-dimensional (associative ) algebras . W e should mentio n th e connection s of that theor y with problems of symmetry in algebra and geometry, including Galoi s theory (th e symmetrie s o f algebrai c equations) , an d wit h problem s o f crystallog - raphy. Eventuall y th e subjec t o f representatio n theor y significantl y expande d i n response t o genera l question s fro m analysis , geometry , an d physics . Fundamenta l discoveries in theoretical physics, such as the theor y o f relativity an d quantu m me - chanics, playe d a significan t rol e i n tha t process . Fo r example , i t turne d ou t tha t logical foundations o f quantum mechanic s ca n b e adequatel y expresse d i n terms of automorphisms o f certai n algebra s (th e algebra s o f observables) . Th e proces s o f describing observable s reduce s t o representatio n theor y o f certai n Li e group s an d algebras. Amon g classica l result s o f that perio d w e specifically mentio n th e work s of E . Car t an an d H . Wey l o n th e genera l aspect s o f the theor y o f Li e group s an d on harmoni c analysi s on compact groups . The underlyin g ide a o f harmoni c analysi s o n group s i s base d o n th e connec - tion betwee n a grou p G an d th e "dua l object " G consisting , roughl y speaking , o f irreducible representation s o f G. Usuall y G ca n b e recovered , u p t o isomorphism , from it s dual object G. A remarkabl e feature o f harmonic analysis is that numerica l functions o n G can be recovered fro m thei r (operator ) "Fourie r images" , where th e role of elementary harmonics is played by irreducible representations of G. A mean - ingful definitio n o f Fourier image s on locally compac t group s i s possible because of the fundamental result s of A. Haar, J . vo n Neumann, an d A . Weil on the existenc e ix
- 15. x PREFAC E (and uniqueness ) o f invarian t measure s o n suc h groups . I n tha t sense , th e classi - cal Fourie r analysi s (Fourie r serie s an d integrals ) i s subsume d int o a n impressiv e development progra m o f harmonic analysi s o n topological groups . Logical foundations o f Fourier analysi s ca n be significantly clarifie d withi n th e framework o f "abstrac t harmoni c analysis" , wher e th e grou p G i s replace d b y a C*-algebra. Fundamenta l result s i n tha t directio n ar e du e t o I . M . Gelfan d an d M. A. Naimark (i n the 1 940s) . Beginnin g with the 1 950s , the theory of C*-algebra s develops ver y rapidl y and , t o a larg e extent , characterize s th e functiona l analysi s of th e 20t h century . I t i s importan t t o observ e tha t tha t theor y ha s fundamenta l applications t o operato r algebras , Hop f algebras , dynamica l systems , statistica l mechanics, quantu m field theory, etc . Modern representatio n theor y deal s with a wide variety of associative algebras , including structure algebra s of manifolds an d Lie groups, universal enveloping alge- bras of Lie algebras, group (convolution ) algebras , Hopf algebras, quantum groups , etc. Notic e that th e theory of Lie groups, born within the context o f differential ge - ometry, is now included in the framework o f functional analysi s by way of bialgebras and forma l group s associate d wit h Li e groups. One ma y als o expand th e definitio n o f representation theor y t o include , i f de- sired, suc h neighborin g discipline s a s abstract theor y o f differential equations , the - ory o f sheaves o n homogenous spaces , microanalysis, quantu m field theory , etc . There i s a know n thesi s accordin g t o whic h "mathematic s i s representatio n theory". Th e correspondin g antithesi s ca n b e state d a s "mathematic s doe s no t reduce to representation theory". I t is worthwhile to note the nature of the question. Whatever i s true , i t appear s tha t th e scop e o f representatio n theor y i s alread y comparable wit h tha t o f the entir e mathematics . It ma y b e that th e desir e to systematize mathematic s i n the spiri t o f represen- tation theor y mad e N . Bourbak i writ e th e multi-volum e se t "Element s o f mathe - matics" . Despit e certai n shortcoming s o f tha t titani c wor k (excessiv e formalism , unfinished parts ) on e finds original treatment o f several fundamental issues , includ- ing general aspects of algebra, topology , the theory of integration, th e theory of Lie groups an d Li e algebras, etc . At present, ther e is a large number o f monographs dealin g with various aspect s of representation theory , includin g Lie groups and Lie algebras ([4] , [10], [14], [31], [35], [61 ]) . Banac h algebra s ([6] , [8] , [1 3] , [22] , [49] , [58]) , algebrai c group s ([3] , [29], [64] , [73]) , infinite-dimensiona l group s ([53]) , genera l representatio n theor y ([40]). Th e author' s monograp h [75 ] ca n b e use d a s a n easil y accessibl e sourc e of informatio n o n representation s o f Li e groups , especiall y suitabl e fo r physicists . However, ther e i s still no monograph whic h would put togethe r al l of those aspect s of representation theory . To fill th e gap , thi s boo k wa s conceive d a s a compilatio n o f canonica l text s on representatio n theory . I t provide s a systemati c descriptio n o f a wid e spectru m of algebro-topologica l structures . O n on e hand , th e concep t o f suc h a boo k i s appealing becaus e i t allow s u s t o compar e idea s an d method s fro m differen t part s of representatio n theory . O n th e othe r hand , i t i s als o risk y jus t becaus e o f th e sheer volum e o f th e materia l t o b e covered . Nevertheless , th e autho r think s tha t a partial resolutio n o f this dilemm a i s possible because th e offere d text s hav e bee n carefully worke d upo n an d refined .
- 16. PREFACE x i The content s o f the boo k spli t int o three parts . Par t I (Introduction ) contain s general fact s fo r beginners , includin g linea r algebr a an d functiona l analysis . Th e survey-type section s o n topology , theor y o f integration , etc . (se e [23] , [24] , [26] , [31]) a s wel l a s Appendice s A , B , C , an d D ar e writte n i n th e sam e spirit . I n the mai n Par t I I (Genera l theory ) w e conside r associativ e algebras , Li e algebras , topological groups , an d Li e groups. W e als o mentio n som e aspect s o f rin g theor y and th e theor y o f algebrai c groups . W e provid e a detaile d accoun t o f classica l results i n those branche s o f mathematics, includin g invarian t integratio n an d Lie' s theory o f connection s betwee n Li e group s an d Li e algebras . I n Par t II I (Specia l topics) w e conside r semisimpl e Li e algebr a an d Li e groups , Banac h algebras , an d quantum groups . The boo k bring s th e reade r clos e t o th e moder n aspect s o f "noncommutativ e analysis", includin g harmoni c analysi s o n locall y compac t groups . Th e autho r regards th e content s o f this book a s a prerequisite fo r thos e who want t o seriousl y study representatio n theory . The styl e o f the boo k allow s the autho r t o choos e th e dept h o f the expositio n to his taste. Fo r example, we prove the theorem o n the conjugac y o f Cart an subal - gebras (i n complex Lie algebras) bu t omi t a similar resul t fo r Bore l subalgebras (i n semisimple Li e algebras) . Ye t th e autho r hope s tha t th e reade r wil l see a detaile d enough panorami c descriptio n o f representation theory . The diverse nature of the compiled materia l unavoidably lead s to discrepancie s in traditions , whic h sometime s caus e certai n redundanc y i n th e definition s an d notation. Fo r example , th e notatio n End X i n th e categor y o f vecto r space s i s sometimes replace d b y L(X) wher e dimX < oo . The exercises included in the book, as a rule, are designed as tests for beginners. Sometimes (i n moderation ) th e result s o f the exercise s ar e use d t o shorte n certai n proofs. Onl y th e exercise s marke d wit h a n asteris k ca n b e viewe d a s mor e o r les s serious problems . While workin g o n th e boo k th e autho r fel t himsel f a chronicler . Indeed , th e book cover s a century i n the developmen t o f mathematics, a period whic h i s prob- ably no t ye t full y appreciated . The content s o f the boo k are , t o a larg e extent , base d o n tw o electiv e course s the author gave at the Independent Universit y of Moscow in 1996-1998. Th e lectur e notes of one of those courses were published i n 2001 ([78]). Th e work on this boo k was partially supporte d b y the RFF I Gran t 01 -01 -0049 0 an d NW O 047-008-009 . The author i s grateful t o V. R. Nigmatullin for his help during the proofreadin g of the text . D. Zhelobenk o
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- 20. CHAPTER 1 Basic Notion s In this chapter we collect basic facts about algebrai c structures, including linear operators o n vecto r spaces . Followin g th e classica l tradition , w e reserve th e ter m linear algebra for the context of finite-dimensional vecto r spaces. Not e however tha t methods o f linear algebr a ar e als o use d i n the stud y o f infinite-dimensiona l vecto r spaces. A s a n example , w e mentio n Sectio n 5 , where , fo r referenc e purposes , w e discuss linear operator s o n Hilbert spaces . 1. Algebrai c structure s 1.1. Semigroup s an d groups . A semigroup i s a n abstrac t se t S togethe r with a given binary operatio n S x S —> 5, (x,y)i — > xy, calle d multiplication, whic h satisfies th e associativity axiom (1.1) x(yz) = (xy)z for al l x,y,z G S. Th e elemen t (1 .1 ) i s denoted xyz. Similarl y (usin g inductio n o n n) on e define s associative words (or monomials) x • • - xn. I n particular , fo r eac h x G S th e associative power x n i s defined . The semigrou p S i s said t o b e commutative i f xy — yxfo r al l x,y G 5, an d S is called a semigroup with identity i f there i s a n elemen t e G S (a n identity ) suc h that (1.2) ex = xe = x for eac h x G S. Th e identit y e is uniquely determine d b y axio m (1 .2) . Indeed , i f e' is another identity , the n e — eef = e'. An element x G S i s said to be invertible i f there i s an element y £ S suc h tha t xy — yx — e. I f we also have xy' = y'x = e for som e y'', then th e equalit y y = ye = y{xy) = (yx)y f = eyf = y' implies th e uniquenes s o f y. Suc h a n elemen t i s denote d x~ l an d i s calle d th e inverse o f x. Not e tha t {xy)'1 =y~ 1 x~ 1 for al l invertible x,y G S. A semigrou p G i s calle d a group i f G i s a semigrou p wit h identit y i n whic h every elemen t i s invertible . A commutativ e grou p i s als o calle d a n abelian group. Sometimes multiplicatio n i n an abelia n grou p i s written additively : (x , y)i — > x 4- y. In tha t cas e th e identit y elemen t i s referre d t o a s th e zero (o r neutral) elemen t OGG. Examples.1 . Fo r eac h se t M le t EndA f b e th e se t o f al l endomorphisms (i.e., transformation s a: M -* M) o f M. I t i s clea r tha t End M i s a semigrou p 3 10.1090/mmono/228/01
- 21. 4 1. BASI C NOTION S with respec t t o th e compositio n o f endomorphism s (1.3) (ab)x = a(bx) where a,b E EndM, x G M. I n thi s example , th e identit y transformatio n ex = x (x G M) i s the identity . 2. Th e subse t Aut M c EndM , consistin g o f automorphisms (i.e. , invertibl e endomorphisms) o f M, i s a group (calle d th e automorphism group of M). The associativit y axio m (1 .1 ) ca n b e viewe d a s a n abstractio n o f th e corre - sponding propert y o f EndM . Examples of abelian groups include: Z (the set of integers), Zp (th e cyclic group of integer s modul o p) , Q (th e se t o f rationa l numbers) . A s example s o f abelia n semigroups we mention Z + (th e set of nonnegative integers ) an d N = Z + {0} (th e natural numbers) . 1.2. Ring s an d fields . A set R wit h tw o associativ e binar y operations , ad- dition (x , y) -+ x + y an d multiplication (x, y)i — > xy, i s called a ring i f R i s a grou p with respec t t o additio n an d th e distributivity condition s hold : (1.4) x(y + z) — xy + xz, (x + y)z = xz + yz for al l x,y,z G R. I n particular , R ha s a zero element 0 such tha t 0 • x = x • 0 = 0 for eac h x G R. The rin g R i s said t o b e commutative (respectively , a ring with identity) i f R is commutativ e (respectively , ha s a n identity ) wit h respec t t o multiplication . A ring wit h identit y i s calle d a skew field i f ever y nonzer o elemen t i s invertible . A commutative ske w field i s called a field. A subse t G C G (respectively , R C R) i s calle d a subgroup of th e grou p G (respectively, a subring o f R) i f G i s close d unde r th e grou p operation s (x,y) i — » xy~1 (respectively , i f R i s close d unde r th e rin g operations) . I n tha t cas e G (respectively, R) i s called a n extension o f G (respectively , o f R). The identit y elemen t o f the rin g R i s usually denote d 1 . W e shall als o assum e that a ring wit h identit y i s nontrivial, i.e. , 0 ^ 1 . Examples. 1 . I f R i s a ring wit h identity , the n R contain s th e subrin g R o f multiples o f the identity , i.e. , 0, ±n, wher e n = 1 + • • • + 1 (n summands) . 2. An y commutativ e rin g R give s ris e t o a n extensio n R[x], consistin g o f al l polynomials i n x. Here , by a polynomial w e understand a n element o f the for m f{x) = a0 + ax H h a nxn , with coefficient s ai G R (i — 0,1,..., n). A mor e detaile d analysi s o f thi s notio n will be given i n 4.1. We also recall the standar d notatio n R , C ,HI (respectively) fo r th e field of real numbers, th e field of complex numbers , an d th e ske w field of real quaternions . 1.3. Vecto r spaces . Give n a field F , a se t X wit h operation s o f additio n X x X— > X, (x , y)H ^ x -f- y an d multiplicatio n F x X— > X, (A , x)i- » Ax, is called a vector space over F i f X i s an abelia n grou p with respec t t o additio n an d (1.5) {x + y) = x + Ay,( A + ji)x = x + fix for al l A , i G F an d x , y G X. I n particular , 0 • x = 0 for eac h x G X, wher e 0 in the left-han d (right-hand ) sid e i s the zer o elemen t o f F {X). Th e element s x G X are calle d vectors, an d th e element s A G F scalars. W e shall cal l the operation s of addition an d multiplicatio n th e vector operations o n X.
- 22. 1. ALGEBRAI C STRUCTURE S 5 We shal l als o us e th e symbo l 0 to denot e a trivial (zero ) vecto r spac e consistin g of a singl e elemen t 0 . Examples. 1 . Th e Cartesia n powe r F n o f any field F ha s a n obviou s structur e of a vecto r spac e ove r F. Indeed , an y x G Fn ca n b e writte n a s a n ordere d n-tupl e X = [X , . . . , X n), where Xi G F (i = 1 , . . . , n) an d th e vecto r operation s o n X n ar e define d compo - nentwise (coordinatewise) . Th e latte r mean s tha t th e vecto r x + y (respectively , Xx) ha s coordinate s xi + yi (respectively , Xxi), wher e i = 1 ,. . . , n. 2. Fo r an y se t M , th e se t F(M) o f al l F- valued function s / : M — » F i s als o a vector spac e ove r F i f th e vecto r operation s ar e define d pointwise . 1.4. Linea r operators . Le t X an d Y b e vecto r space s ove r a field F. A ma p a: X — > Y i s sai d t o b e linear i f (1.6) a(A x + iiy) = Xax + iay for al l A,/ i G F an d x,y e X. I n particular , a(0 ) = 0 (wher e th e sam e symbo l 0 denotes th e zer o element s i n X an d Y). Linear map s a : X — » Y ar e als o called linear operators (fro m X t o Y ) o r homo- morphisms (fro m X t o Y) . Th e latte r ter m emphasize s th e fac t tha t operator s (1 .6 ) respect th e vecto r operation s i n X an d Y . The se t o f al l homomorphism s a: X — » Y i s denote d Hor n (X, Y). I t i s clea r that Hor n (X, Y) i s a vecto r spac e ove r F wit h respec t t o th e operation s (1.7) (Xa +[ib)x = Xax + fibx, where A,^ x G F an d a,b <E Hor n (X, Y), x G X . Linea r operator s a : X — > • X ar e called endomorphisms o f X , an d thei r totalit y i s denote d E n d X = Hor n (X, X) . We remar k tha t th e symbo l E n d X i s onl y use d fo r th e se t o f linear map s (1 .6) . Sometimes (t o avoi d possibl e confusion ) instea d o f E n d X on e use s th e symbo l L(X). There i s an importan t generalizatio n o f the notio n o f linear map . Namely , give n any collectio n o f vecto r space s Xi (i = 0 , 1 , . . . , n) , a ma p a : X x • • • x X n —* X Q is sai d t o b e multilinear (o r n-linear ) i f i t i s linea r i n eac h argumen t Xi G X % [i — 1,.. . , n). I f n = 2 , 3 on e use s th e term s (respectively ) bilinea r an d trilinear . 1.5. Algebras . A vecto r spac e A ove r a field F i s calle d a n algebra (ove r F) if it carrie s a n F-bilinea r operation , calle d multiplication, A x A — > • A: (x , y)i — > xy. The bilinearit y conditio n i s expresse d b y th e distributivit y condition s (1 .6 ) an d (1.7) wher e w e assum e tha t a , b,x,y G A. The algebr a A i s sai d t o b e associative (respectively , commutative) i f th e mul - tiplication o n A i s associativ e (respectively , commutative) . Th e algebr a A i s sai d to b e unital (o r an algebra with identity) i f i t ha s a multiplicativ e identity , whic h is usuall y denote d 1 . A subse t X C X (respectively , A C A) i s called a subspace o f X (respectively , a subalgebra o f A) i f Xi (respectively , A{) i s closed wit h respec t t o th e vecto r oper - ations i n X (respectively , algebrai c operation s o n A). I n tha t cas e X (respectively , A) i s calle d a n extension o f X i (respectively , o f A). If A i s a unita l algebra , the n th e ma p F —> A, A i —> A • 1 is a n embeddin g o f F as a subalgebr a o f A. Thu s w e ca n writ e A G A instea d o f A • 1 G A.
- 23. 6 1. BASI C NOTION S Examples. 1 . Fo r each vector space X ove r F th e set L(X) = End X (se e 1 .4 ) is an associativ e algebr a ove r F. 2. Th e algebr a L(n ) = Ma t (n,F) consistin g o f n x n-matrices wit h entrie s i n F i s an associativ e algebr a ove r F. 3. Th e subalgebra D(n) C L(n) of all diagonal matrices in L(n) is commutative. 1.6. Algebrai c structures . A n algebraic structure i s a se t S togethe r wit h a collection of n-ary relation s (i n the Cartesia n power s S n ) an d a system of axioms for thos e relations . Usuall y thos e relation s ar e writte n a s functions , i.e. , map s Sn — > 5m . A s example s o f such map s w e mentio n th e operation s o f additio n an d multiplication considere d above . In tha t sense , al l example s considere d s o fa r (semigroups , groups , rings , etc. ) are special cases of algebraic structures. Tw o such structures Si an d 5 2 are said t o be of the same kind i f they ar e define d b y the sam e relations an d axioms . The convenienc e of using the same terminology fo r various algebraic structure s leads us to the concept o f a morphism o f algebraic structures. Namely , a morphis m from a structur e Si t o a structur e 5 2 o f th e sam e kin d i s an y ma p (p: Si— » 52 respecting thes e structure s (thi s mean s tha t ip maps th e structura l relation s i n Si to th e correspondin g relation s i n 52). For example , a morphis m (o r homomorphism ) betwee n semigroup s Si an d 5 2 is any ma p <p>: Si — » 52 preserving th e operatio n o f multiplication, i.e. , (1.8) (p(xy) = <p(x)<p(y) for al l x,y G Si. I f Si an d 5 2 ar e semigroup s wit h identity , the n w e assume , i n addition, tha t (1.9) <p(e) = e, where e i n th e left-han d (right-hand ) sid e o f thi s equalit y i s th e identit y o f 5 i (respectively, 52) . If Si an d 5 2 are groups, then (1 .8 ) implies (1 .9) . Thi s follows from th e identit y (p{e)2 = (p(e) (sinc e 99(e) -1 i s defined) . Moreover , i t follow s fro m (1 .8) , assumin g xy — e, tha t (1.10) ^(x)" 1 = ^(x" 1 ) for eac h x G Si. Similarly on e ca n conside r morphism s o f rings , vecto r spaces , algebras , etc . The set of all morphisms</? : 5i— » £2 is denoted Mo r (5i, 52) or Hom (5i, 52). Tw o structures Si an d 5 2 o f th e sam e kin d ar e sai d t o b e isomorphic i f there exist s a morphism ip: Si —> S2 admittin g a n invers e Lp~ l : S2— » Si. I n tha t cas e on e use s the notatio n 5 i « 52 - Examples.1 . Th e exponentia l functio n y — ex i s a n isomorphis m betwee n the additiv e grou p o f R an d th e multiplicativ e grou p Ri — (0, +00). 2. A n isomorphis m ip G Hom (5, 5) i s calle d a n automorphism o f th e struc - ture 5 . Th e se t o f all such automorphism s i s denoted Au t 5. I t i s clear tha t Au t 5 is a group (wit h respec t t o the compositio n o f morphisms define d i n Hom (5, 5)). 1.7. Categories . Logica l difficulties i n set theory prevent u s from formin g th e set of all sets, eve n fo r th e set s wit h a give n algebrai c structure . Instea d w e for m classes of sets. Fo r example, the statement tha t G is a group can als o be expresse d by saying that G belongs to the class of groups.
- 24. 1. ALGEBRAI C STRUCTURE S 7 A class of sets K i s called a category if for an y two sets A an d B fro m i f (suc h sets ar e calle d objects of the category if ) a set of morphisms Mo r (A, B) i s given , subject t o th e followin g axioms : (a) I f eithe r A ^ A' o r B / £' , the n Mor (A, 5) an d Mor {A', B') hav e a n empty intersection . (/?) Fo r an y thre e object s A, B,C o f i f a n associativ e compositio n la w i s de- fined: (1.11) Mo r {A, 5) • Mor (B, (7 ) C Mor {A, C). (7) Fo r an y objec t A o f i f ther e i s a morphis m 1 ^ G Mor (A, A) tha t act s identically o n the lef t an d o n the righ t i n (1 .1 1 ) . As in 1.1 , it is easy to check that 1 ^ is unique. Ofte n th e set Mor (A, B) consist s of maps <p: A— > B an d compositio n (1 .1 1 ) coincide s with the usua l compositio n of maps. Fo r al l algebrai c structure s considere d abov e (se e 1 .6 ) w e us e th e notatio n Horn (A, B) rathe r tha n Mo r (A, B). Examples. 1 . Th e category of sets, denote d SET , is defined b y maps (p: A—* B, wher e A an d B ar e arbitrar y sets . 2. Th e category of groups, denote d GROUP , i s define d b y grou p homomor - phisms ip: G—• H. 3. Th e category of vector spaces over a field F i s denoted VEC T (= VECT^?) . In this category , Mo r (X, Y) = Horn (X, Y) i n the notatio n o f 1 .4 . The reader ca n now enlarge this list to include semigroups, rings, algebras over a field, etc. Examples. 1 . Fo r eac h operator a G Horn (X, Y ) i n VECT^? it s kernel ker a = {x G X: ax — 0} and it s image ima = aX ar e subspaces (respectively ) i n X an d Y. 2. Similarly , ker a and im a in the category ALG^ are subalgebras (respectively ) of X an d Y. 1.8. Categorica l vocabulary . T o compar e differen t categorie s on e use s a special vocabular y tha t include s suc h notion s a s a subcategory, th e dual category, covariant (o r contravariant) functors betwee n categories , etc . For example , a subcategor y if i o f th e categor y i f i s define d a s a subclas s if 1 C if (endowe d wit h th e sam e morphism s a s if) . Th e categor y if op dua l t o i f has the sam e objects a s if bu t eac h Mor (A, B) i s replaced wit h Mor (B, A). We say that a map <£ : if 1—> if2 fro m a category i f 1 to a category if 2 i s given if to each object A i n ifi ther e corresponds a n object A! = $(A) i n if2 an d to each morphism fro m Mo r (A, B) i n ifi ther e correspond s a morphism fro m Mo r [A', B') in if2 . Th e ma p< £ > : ifi— » if 2 i s calle d a covariant functor (fro m if i t o if2 ) i f i t preserves al l compositions (1 .1 1 ) . A covariant functo r 3> : ifi— * K^ 9 i s also calle d a contravariant functor K — » if2. A covarian t functo r<£ : if 1—> if 2 i s calle d a n isomorphism (equivalence) be - tween if i an d if 2 i f it admit s a n inverse functor <I> - 1 : if2— > if i define d u p t o a n isomorphism (i.e. , ^> _1 (^>(A)) « A fo r al l object s A o f if i an d ^(^~ 1 (B)) « B for al l object s B o f if2) - A covariant functo r< £ > : ifi— » if 2 i s called a n embedding of if i i n if 2 i f $ give s ris e t o a n isomorphis m betwee n if i an d som e subcategor y of if 2.
- 25. 8 1. BASI C NOTION S We shal l occasionall y us e th e categorica l vocabular y t o shorte n som e state - ments. Sometime s (i n Part III ) suc h shortenings wil l be substantial . Example. Fo r eac h morphis m a: M — » N i n SE T w e ca n defin e th e dua l morphism a * : F(N) - • F(M) (se e 1 .3 ) b y (1.12) (a*f)(x) = f(ax), where / G F(N) an d x G M. I t i s eas y t o chec k tha t (ab)* = 6*a * fo r composi - tions (1 .1 1 ) . Thus th e ma p Mi — > F(M) give s ris e to a contravarian t functo r fro m SE T t o the categor y o f algebras ove r F. 2. Vecto r space s 2.1. Notation . Le t X b e a vecto r spac e ove r a field F. W e shal l conside r systems of elements e = (e^)^ / o f X (wher e / i s an arbitrar y se t o f indices). Finit e sums (2.1) x = J2 X ^ whereA^ G F (wit h only finitely man y A ; ^ 0 ) are called linear combinations o f the ei (i e I). Th e se t Fe o f al l suc h linea r combination s i s called th e linear hull (or span) o f e. Clearly, Fe i s a subspac e o f X. I f X — Fe, the n w e shal l sa y tha t X is spanned b y e . A system e is said t o b e linearly independent (respectively , spanning) i f x = 0 in (2.1 ) implie s tha t A ; = 0 for eac h i G I (respectively , i f F e = X). Linea r inde - pendence i s equivalent t o the uniquenes s o f decomposition (2.1 ) . I n that cas e (2.1 ) is written a s yz.z) x — y x^e^, i where th e Xi (i G I) ar e uniquel y determine d b y x. Her e w e set Xi — Si(x). Th e coefficients xi (i G I) ar e calle d th e coordinates o f x (wit h respec t t o e) . Each linearl y independen t spannin g syste m e in X i s called a basis of X. An y vector x G X i s the n uniquel y represente d i n th e for m (2.2) , wit h coordinate s %i ~ &i %) • Examples. 1 . Th e coordinat e spac e Fn (se e 1 .3 ) ha s a basis consisting o f th e elementse ^ = (0,... , 0,1, 0,..., 0) wit h 1 at it h place . I n thi s cas e th e notatio n x — (xi,..., xn) identifie s th e coordinate s X{ — £i(x) o f x G Fn wit h respect t o th e basis (ei), wher e i — 1,..., n. 2. Le t F[M] C F(M) b e the subse t o f all finite function s f:M->F, i.e. , th e functions whic h ar e no t zer o only a t a finite numbe r o f points o f M. Clearl y F[M] is a vecto r subspac e o f F(M) wit h "delta-functions " S a (a G M) a s a basis, wher e da(x) = 0 when x ^ a and S a(a) = 1 . Henceforth w e shall identif y a G M wit h th e delta-functio n 5 a G F[M. A s a result, M embed s (a s a basis) i n F[M. Th e space F[M] i s called th e formal linear hull (ove r F) o f M.
- 26. 2. VECTO R SPACE S 9 2.2. Bases . Accordin g to the well-known theorem of Hamel (see , for example , [1]), every vector space X over a field F has a basis. Th e proo f o f this theore m i s based o n the Zor n lemma , whic h implie s th e existenc e o f a maximal (wit h respec t to inclusion ) linearl y independen t syste m i n X. A s a n eas y exercise , th e reade r should chec k that eac h such syste m i s a basis of X. A substantial refinemen t o f Hamel's theore m i s that any two bases of X are of the same cardinality. W e shall sketch a proof o f this statement . If X admit s a finite basis, then th e proof i s an easy exercise. Assum e now tha t X ha s base s A an d B wit h infinit e cardina l number s a = card A an d (3 — card B. Note that A is a disjoint unio n of subsets An ( n G N), where An consist s of elements a G A whic h ca n be represented a s a linear combination , wit h nonzer o coefficients , of exactly n element s b G B. I t i s easy t o chec k that car d .An < nf3 n . Therefor e oo (2.3) a < ]Tn/T. 7 1 = 1 It i s well known (see , for example , [65] ) that (3 2 — (3 for eac h infinite cardina l f3, so that j3 n — (3 for eac h n G N. Moreover , th e right-han d sid e of (2.3 ) coincide s wit h j3 and therefor e a < (3. Similarly, (3 < a an d therefor e a — j3. Thus every vector space X give s rise to a unique cardinal number dimX , calle d the dimension o f X, an d define d a s the cardinalit y o f an arbitrar y basi s of X. Th e space X i s said to he finite-dimensional (respectively , infinite-dimensional) i f dim X is finite (respectively , infinite) . Examples. 1 . dimF n = n. 2. Ther e ar e vecto r space s o f arbitrar y dimension . Fo r example , dimF[M ] = cardM. Exercise. Chec k (similarly to the proof of Hamel's theorem) that ever y linearly independent syste m i n a vecto r spac e X i s containe d i n som e basi s o f X. Fo r example, eac h nonzer o vector i s part o f a basis of X. 2.3. Proposition . Let a G Horn (X, Y) (znVECT F). Then (a) For each basis e^ (i G /) of X, the operator a is uniquely determined by its values aci. (/?) For each collection f % G Y (i G I), there is an operator a G Horn (X, Y) such that aei — fi for all i G /. PROOF. Fo r the first assertion, it suffices t o notice that, in the notation of (2.2), (2.4) ax — y^ jXja(el). i For th e secon d assertion , on e quickl y check s tha t th e sam e formul a define s a as a linear map . • As a consequence, w e have the rul e (2.5) H o m ( X , Y ) ^ y a , where a = dimX . 2.4. Corollary . The equality dim X = dim F for X and Y in VECT^ ? is equivalent to X « Y.
- 27. 10 1. BASI C NOTION S Indeed, i f d i m X = di m Y, the n th e formul a ae % — fi, wher e a an d fa ar e base s (respectively) i n X an d Y , define s th e desire d isomorphis m X « Y . Examples.1 . Eac h n-dimensiona l vecto r spac e X (n G Z+ ) i s isomorphi c t o Fn . 2. Eac h vecto r spac e X ove r a field F i s isomorphi c t o F[M] , wher e d i m X = cardM. 2.5. Matrices . Le t e = (cj) jej an d / = {fi)iei b e base s i n vecto r space s X and (respectively ) Y . Expressin g ae 3 (j G J) i n term s o f th e element s o f / , w e hav e (2.6) ae 3 = ^ P a ^ / i * i with a tJ G F. The collectio n a/ 5e = (a^) , wher e i G / an d j G J, i s calle d th e matrix o f a (i n the base s / an d e) . Th e element s a ^ wit h fixed i (respectively , j) ar e calle d th e rows (respectively , columns) o f th e matri x a/ }€>. Accordin g t o (2.6) , th e column s aej o f aj 5 e satisf y th e followin g finiteness condition : ($) Fo r eac/ i j E J only finitely many numbers a^ are different from zero. Conversely, i t follow s fro m 2. 3 (/3 ) tha t eac h matri x a^ e satisfyin g conditio n (<£) defines a uniqu e operato r a G Horn (X, Y) wit h a/ ?e a s it s matri x (wit h respec t to / an d e) . A s a consequence , th e ma p a -* af^ e i s a n isomorphis m betwee n Hom(X, Y ) an d th e vecto r spac e o f al l matrice s (2.6 ) satisfyin g conditio n ($) . If th e base s / an d e ar e fixed, w e shal l simpl y writ e a instea d o f a/ >e (i n othe r words, w e shal l identif y th e operato r a G Hor n (X, Y) wit h it s matri x (2.6)) . A s an exercise , th e reade r shoul d chec k tha t th e compositio n o f operator s a — be (whenever i t i s defined ) correspond s t o th e standar d matri x multiplicatio n (2.7) a,ij = }^b ikckj. k In particular , th e algebr a E n d X i s isomorphi c t o th e matri x algebr a M a t ( n , F ) , where n = dimX , consistin g o f al l squar e matrice s o f orde r n satisfyin g condi - tion(<£>) . Normally, w e shal l us e th e symbo l Ma t (n , F) whe n n < oo . I n tha t cas e Mat (n , F) consist s o f al l n x n matrice s ove r F. 2.6. Dua l spaces . A vecto r spac e X * consistin g o f al l F- valued linea r func - tions (als o calle d functionals ) o n X i s calle d th e (algebraically ) dual space o f X (X* = H o m ( X , F ) ) . As i n (2.5) , th e value s (2.8) f(x) = Y,fi x *> where Xi — £i(x) ar e th e coordinate s o f x i n th e basi s e (se e 2.1 ) , o f eac h functiona l / G X* ar e uniquel y determine d b y th e coefficient s f t G F. I n tha t sense , (2.8 ) ca n be writte n a s (2.9) X * ^ F a , where a = d i m X .
- 28. 2. VECTO R SPACE S 11 Let e — (ei)iei b e a linearl y independen t se t i n X. W e wan t t o sho w that , in X*, ther e exist s a dual system Si (i G I) give n by (2.10) £i(e j) = 5 ij, where 6ij i s Kronecker' s delt a (5 i3 — 0 when i ^ j , and Su = 1 ) . Indeed , e ca n be extended t o a basis of X (se e 2.2). No w define Si(x) = Xi as the coordinate s of x G X i n that basi s (se e 2.1). In particular , fo r eac h nonzer o x G X ther e i s an / G X* suc h that f(x) ^ 0 . The dual systems allow us to find explicit expressions for matri x elements (2.6 ) of a. Namely , (2.11) dij = ipi(aej), where ipi (i G I) i s the syste m dua l t o the basi s fi (i G I)o f Y. Examples. 1 . Th e vecto r spac e F n ca n b e identifie d wit h it s dua l vi a (2.8) , where i = 1 ,.. . , n. 2. Th e spac e dua l t o F[M] ca n be identifie d wit h F(M) usin g the formul a 9(f) = £/(*)<?(*), X where / G F[M) an d # G F(M). 2.7. Bilinea r forms . A bilinear for m f: X xY -+ Z (in VECTF) i s said t o be nondegenerate i f for eac h 0 / x G l (respectively , 0 ^ y G Y) ther e i s a 7 / G Y (respectively, x G l ) suc h that /(# , y) ^ 0 . The nondegenerac y o f / i s equivalen t t o th e injectivit y o f eac h o f th e map s x» —> f x an d y *-* fy, wher e (2.12) fx(v) = f(x,y) = f y(x). In thi s sens e (2.1 2 ) give s ris e t o embedding s X (respectively , Y ) i n Hor n (Y, Z) (respectively, Hor n (X, Z)). In general , th e left kernel ker ^ / (respectively , th e right kernel ker p /) o f / is define d a s th e kerne l o f th e ma p x i — > fx (respectively , y -^ f y). Thu s / i s nondegenerate mean s that ker> , / = ker p / = 0 . Examples.1 . I f Z = F an d / i s nondegenerate , the n (2.1 2 ) give s ris e t o embeddings X - • Y * an d Y -> X* . 2. Th e bilinea r for m (x,?/ ) = y(x ) i s nondegenerate o n X x X * (se e 2.6) . I n this cas e (2.1 2 ) become s (2.13) x(y) = (x,y)=y(x), so that th e first equalit y give s ris e t o a n embeddin g o f X int o th e bidua l X* * = (X*)*. 3. I f dimX < oo , then thi s embeddin g i s an isomorphism , X w X**. 4. Formul a (2.1 1 ) ca n als o be written a s (2.14) dij = (aej,(pi). The bilinea r for m (2.1 3 ) i s called th e canonical bilinear form o n X x X* . Exercise. Le t (• , •) be a nondegenerat e symmetri c bilinea r for m o n a finite- dimensional vecto r spac e X. Th e relation s x(y) = (x,y) = y{x)
- 29. 12 1. BASI C NOTION S give rise t o a n isomorphis m X « X*. Th e symmetr y propert y o f (• , •) means tha t (x, y) = (y , x) fo r al l x an d y £ X. 2.8. Adjoin t operators . Fo r eac h operato r a G Horn (X, Y) w e defin e th e adjoint operator a * G Hom(Y*,X*) b y (2.15) (ax,y) = (x,a*y) (in accordanc e wit h (1 .1 2)) . Indeed , fo r eac h y G Y*, relatio n (2.1 5 ) determine s a*y e X* uniquely . Note tha t th e ma p a i — > a* i s linea r an d tha t (afr) * = b* a* (whenever th e composition a 6 is defined). I f dimX, di m Y < oo , then a* * = a . Sometimes i n thi s situatio n th e notatio n a »-» a* i s replaced b y eithe r a ^ a' or a ^ a * (transpose) . Exercises. 1 . Th e matrice s o f a and a 7 i n the base s e and / (in , respectively , X, Y) an d th e dua l base s e,(p (in , respectively , X* , Y*) ar e th e transpose s o f each other, i.e . (2.16) a' %3 = a^ . 2. Replacin g th e base s e and / b y new base s (2.17) ej =^2u 3jes, ft = ^v rlfr, s r where u = (u SJ) an d v = (v ri) ar e invertibl e matrices , change s a t o a new matri x (2.18) a = v~ 1 au. In particular , whe n X = Y an d e = / , th e matrice s a an d a ar e similar, i.e. , a = u~ 1 au. 2.9. Direc t sums . Le t X2 (i G /) b e an arbitrary famil y o f vector spaces over a field F. Th e direct sum (2.19) X = ® X * i is defined a s a subspace o f the Cartesia n produc t (2.20) X = Y[Xt, i consisting of all finite sequence s x — (x^)2Gj, wher e Xi G Xt (i G I). Her e the vector operations x + y an d Xx ar e define d componentwise . A vector x G X i s said t o b e /mz£e i f x2 ^ 0 only fo r finitel y man y x % (i G I). In tha t cas e eac h operato r a G End X ca n b e describe d b y a block matrix a = {p>ij), wher e (2.21) a^ = pzapj G Horn (Xj, Xi), and pi i s the projectio n operato r i n (2.1 9 ) ont o Xi. Her e on e view s a^ a s its ow n restriction t o Xj. It i s easy to check that bloc k matrices (2.21 ) compose according to (2.7 ) (when - ever th e compositio n ab is defined) .
- 30. 2. VECTO R SPACE S 13 An operato r a G En d A i s sai d t o b e block-diagonal (relativ e t o decomposi - tion (2.1 9) ) i f dij = 0 whe n i ^ j . I n thi s cas e w e writ e a = 0 z a % or a = dia g (a^) , where ai — a^. I f / = { 1 , . . . , n} , then w e writ e n (2.22) a = ££)a l = d i a g ( a i , . . . , a n ) . Exercise. Linea r functiona l / G A* o n (2.1 9 ) ar e o f th e for m (2.23) f(x) = Y^fi(xi), i where th e f t G A* (i G /) coul d b e arbitrar y linea r functionals . Therefor e (2.24) A * = []X * i Note tha t (2.23 ) i s a finit e su m (sinc e an y x G A i s finite) . 2.10. Quotien t spaces . Le t A Q be a subspac e o f a vecto r spac e A . Settin g x ~ y whe n x — y G AQ , we hav e a n equivalenc e relatio n o n A wit h equivalenc e classes (2.25) IT(X)=X + X Q, where x G A (i.e. , 7r(x ) i s th e se t o f al l vector s x + y, wher e y G A0 ). Th e relatio n TT(X) = n(y) i s equivalen t to x ^ y. I t i s no w clea r tha t th e quantitie s define d b y (2.26) XTT(X) — 7r(Ax), 7r(:r ) + 7r(y ) = 7r( x + y) do no t depen d o n th e choice s o f x an d y i n th e classe s 7r(x ) an d rc(y). Th e ne w vector spac e 7r(A ) i s denote d A/A o an d calle d th e quotient space o f A b y th e subspace XQ. Note tha t (2.26 ) mean s tha t ix X —> X / A Q i s linear . Th e ma p TT i s calle d th e canonical projection (o f A ont o A/Ao) . Choosing a basi s e o i n A o an d extendin g i t t o a basi s eoUe i i n A w e hav e (2.27) A = A 0 © Ai, wher e X 1 = Fe x « A / A 0 . Therefore di m A = di m Ao + di m X. Th e secon d summan d i s calle d th e codimen- sion o f A Q i n A an d i s denote d codi m XQ. Example. I f codi m AQ = 1 , the n A 0 i s a hyperplane i n A determine d b y th e linear equatio n TT(X) — 0. Accordingly , (2.27 ) become s (2.28) A = A 0 © F e 0 , for an y e o no t containe d i n AQ . I t i s als o clea r tha t classe s (2.25 ) coincid e wit h hyperplanes TT(X) = const . 2.11. Invarian t subspaces . A subspac e A Q C A i s said t o b e invariant wit h respect t o a G EndA " i f aA' o C A' o (i.e . ax G A^Q for eac h x G Ao). I n tha t cas e th e relation (2.29) 7r(a)7r(x ) = 7r(ax) gives ris e t o a n induced operator a = re {a) G End A, wher e A = A / A 0 . If w e represen t operator s a G En d A a s bloc k matrice s a — (a 2J), wher e z,j = 1 , 2 (i n accordanc e wit h (2.27)) , the n th e invarianc e conditio n aA 0 C A 0 is equivalen t t o th e conditio n a^ = 0 (i.e. , t o a bein g uppe r triangular) .
- 31. 14 1. BASI C NOTION S Exercise. Eac h operato r a G Horn (X, Y) give s rise t o a vector spac e isomor - phism (2.30) X/kera^ima. 2.12. Th e groun d field . Late r i n thi s boo k w e will hav e t o impos e certai n conditions o n the groun d field F. The field F is said to b e algebraically closed if F contains th e root s o f all alge- braic equations f(x) — 0, where / i s a polynomial with coefficients i n F. I n general, F has an algebraic closure, i.e., an algebraically closed extension F (meaning that F is embedded i n F as a subfield). See , for example , [1 ] . Her e is a classical example : C i s an algebrai c closur e of R . We say that F is a field of characteristic p — 0 (respectively, p ^0 ) if n ^ 0 for each n G N (respectively , p is the smalles t o f the number s n G N such tha t n — 0). Here, by definition, n = H h i (see 1.2). Th e characteristic of F i s denoted p = charF. I f p / 0 , then p is a prime numbe r (exercise) . Her e is an example : th e residue field Fp = Z/pZ. Sometimes we shall consider generalizations of vector spaces (and, in particular, of algebras) wher e th e groun d field F is replaced wit h a ring R. For example, the matrix algebr a Mat (n, R) ove r a ring R, wher e n is a cardinal number, i s defined a s the algebr a o f al l matrice s wit h element s a^ G R(z, jG J), where card / = n, satisfyin g conditio n ($ ) fro m 2.5 . Exercises. 1 . Prove tha t Hamel' s theore m generalize s t o vecto r space s ove r skew fields. 2*. Tr y to prove the Frobenius theorem: eac h finite-dimensional ske w field over R i s isomorphic t o on e of the ske w fields R, C, H. 3. I n particular , eac h finite-dimensional field ove r R is isomorphic t o eithe r R or C . Se e also [1]. 3. Element s o f linea r algebr a 3.1. Determinants . Throughou t thi s sectio n X wil l b e a finite-dimensional vector spac e ove r a n algebraicall y close d field F. A s is well known , i n this case matrix theor y become s a n effectiv e too l fo r studyin g linea r operator s a G EndX. First not e tha t th e notio n o f determinan t extend s fro m th e matri x a e = a ee (see 2.5 ) t o th e operato r a G EndX. Indeed , fo r eac h pai r o f base s e and / o f X the matrices ae an d aj are similar (se e 2.8), whence detae = det ay. Therefore , th e equality (3.1) de t a = det ae defines a unique numbe r de t a, calle d th e determinant o f a. Th e languag e o f ma - trices ca n b e use d t o stat e th e well-know n criterio n fo r invertibilit y i n EndX : an operator a G EndX i s invertible if and onl y i f det a ^ 0. An elemen t a G Fis called a n eigenvalue o f a if there exist s a nonzero vecto r (an eigenvector) x G Xsuch tha t ax = ax, i.e. , (a — a)x = 0. Suc h a vector exist s if and onl y i f the operato r a — a is not invertible , i.e. , (3.2) de t (a - a) = 0. Here we identify a G Fwith th e operato r a • 1 G EndX ( 1 is the identit y operato r in EndX) . Sinc e F is algebraically close d an d th e left-han d sid e o f (3.2) i s a
- 32. 3. ELEMENT S O F LINEA R ALGEBR A 15 polynomial i n a, we conclude tha t (3.2 ) i s solvable, i.e. , eac h operato r a G EndX has a t leas t on e eigenvalue a G F. This show s tha t th e se t o f eigenvalues o f acoincides wit h th e se t o f roots (disregarding th e multiplicity ) o f characteristic equation (3.2) . The numerical function pa (A) = det(A—a) is called the characteristic polynomial of a. Not e tha t n (3.3) Pa(A ) = $ > i ( a ) A S where a^(a ) ar e polynomial s i n a (i.e., polynomial s i n th e matri x element s a). In particular, a n(a) = 1 and cro(a) = (—l)n deta. Her e n — dimX. 3.2. Invarian t flags. A chain of subspaces T 0 = X0 c X C • • • C Xn = X is called a flag of subspaces of X if dim(AQ/X^_i) = 1 for eac h i = 1 ,... , n, s o tha t n = dimX. A flag T is said to be invariant wit h respect t o an operator a G EndX if each subspac e Xi is invariant, i.e. , aX % C Xi for eac h i — 1,..., n. Define a basise ^ (i = 1,..., n) o f X by th e conditio n Xi = X^_ i 0 Fe^. Th e invariance o f T means tha t aei = a^ei (mo d Xi-i) for eac h i = 1 ,... , n. I n othe r words, the matri x o f a in the basi s just define d i s upper-triangular : (Oil * OL2 (3.4) 0 a n / Here th e sta r stand s fo r a collection o f matrix element s a ^ (1 < i < j < n). Conversely, i f the matri x o f a i n som e basi s is of th e for m (3.4) , the n thi s basi s gives ris e t o a n a-invarian t flag . Usin g thi s basi s to compute th e characteristi c polynomial o f a, w e have n (3.5) VaW^W^-cxi). 2 = 1 3.3. Theorem . Any operator a G EndX admits an invariant flag, i.e., in some basis itcan be made upper-triangular. Moreover, for any ordering of the roots ai of the polynomial (3.5 ) there exists a triangular matrix (3.4 ) for a. PROOF. Le t e be a n eigenvecto r o f acorresponding t o the eigenvalu e OL. Making e par t o f a basi se ^ (i = 1,..., n) o f X we see that, i n thi s basis , a ha s a block-triangular matri x I Oil * (3.6) a = ( Q fli j, where a i s a square matrix of order n — 1. Computin g the characteristic polynomia l pa i n this basis we have Pa(A) = (A-ai)p a i (A), so that th e root s o f pai ar e obtained fro m th e root s of pa b y removing the roo t ai . Now we can appl y a n inductio n argumen t (o n n = dimX ) t o th e operato r a. As a result , matri x (3.6 ) become s matri x (3.4) . •
- 33. 16 1. BASI C NOTION S Example. I f a has a unique eigenvalu e a , the n (3.7) p a(A) = ( A - a) n , i.e., a i s of multiplicity n. Exercise. Matri x a i n (3.6 ) i s the matri x o f the operato r a = n{a) induce d by a (se e 2.10) o n the quotien t spac e X = XjFe. Applying Theore m 3. 3 we have the followin g importan t resul t abou t th e sepa- ration of the eigenvalues o f a G EndAC. 3.4. Theorem . Let X t {i — 1,..., k) be the set of all pairwise distinct eigen- values of a G EndX. Then a is the direct sum of operators a^ (i = 1 ,... , k), where a{ has a unique eigenvalue A?. . In other words, X is the direct sum of a-invariant subspaces k (3.8) X = Q)X U i=l where ai = axi ( i = 1 ,... , &). PROOF. B y Theorem 3.3 , the operator a has, in some basis, a block-triangula r matrix (3.9) a = ( j where a ha s a unique eigenvalu e A i an d th e eigenvalue s o f 5 are differen t fro m Ai . Replacing a by a — Ai , w e ma y assum e withou t los s o f generalit y tha t A i = 0 , s o that a i s nilpotent (i.e. , am = 0 for some m) an d that 5 is invertible. W e shall now show that, unde r th e ne w assumption , th e matri x equatio n 'l x fa 0 _ fa 0 f x^ can b e solve d i n x, s o that a is similar t o a$ = dia g (a , S). Note tha t th e l' s i n (3.1 0 ) stan d fo r th e identit y operator s o f the appropriat e orders. Equatio n (3.1 0 ) reduce s t o (3.11) xS — ax = j3. Setting oo (3.12) x = ^2a k p5~k - 1 , k=0 we have a finite su m (becaus e a 1 71 = 0) . Not e tha t x5 differ s fro m ax onl y i n th e term correspondin g t o k = 0, whence (3.1 1 ) . Thus (3.1 2 ) i s a solution o f (3.1 1 ) . Applyin g no w induction on n = dimX t o 5 we have desire d decompositio n (3.8) , i.e., a = diag (ai,... , a^). • Example. I f all the roots of the characteristic polynomial pa ar e of multiplicity one (i.e. , a ha s n distinc t eigenvalues) , then , i n som e basis , a diagonalizes : a — diag ( a i , . . . , a n ) . 3.5. Corollary . Each operator a G EndX satisfies the Cay ley-Hamiltonequa- tion (3.13) Pa(a)=0 ,
- 34. 3. ELEMENT S O F LINEA R ALGEBR A 17 where the left-hand side denotes the result of the substitution A n a in p a. Indeed, i f a ha s a unique eigenvalu e a , the n Theore m 3. 4 show s tha t (3.14) (a-a) n = 0 . In the genera l case , Theorem 3. 4 implie s tha t k i= where mi — dimX^, whic h yields (3.1 3) . Exercise. Sho w that fo r a given operator a G End X th e followin g condition s are equivalent : (a) (a — a)n = 0, where n = dimX ; (/?) fo r eac h x e X ther e i s a k — k(x) suc h that ( a — a)k x — 0; (7) a ha s a unique eigenvalu e a ; (5) th e characteristi c polynomia l pa i s of the for m (3.7) . 3.6. Roo t subspaces . Theore m 3. 4 would b e mor e complet e i f we explicitl y described th e subspace s Xi (i = 1 ,... , k). For eac h A G F le t X(a) b e the se t o f all vectors x e X satisfyin g (3.15) (a-X) k x = 0 whe n k>k 0(x). Obviously X(a) i s an a-invarian t subspac e o f X. I t i s called th e root subspace of a corresponding t o th e root A. It i s clea r tha t X(a) contain s th e eigenspace ker( a — A) of a spanne d b y th e eigenvectors o f a correspondin g t o th e eigenvalu e A . Moreover , X(a) 7 ^ 0 onl y when ker( a — A)7^ 0, i.e., when A i s an eigenvalue o f a. I t i s also clear tha t a has a unique eigenvalu e A i n X(a). The subspac e X 0(a) i s called th e nil-space o f a. 3.7. Theorem . X is a direct sum of the root subspaces of a : (3.16) X = 0XA (a), A where X(a) 7 ^ 0 only when A = A ^ (i = l,...,n ) in the notation of 3.4 , so that (3.1 6 ) coincides with (3.8) . PROOF. A S wa s show n i n 3.4 , eac h a G EndX i s a direc t su m a$ © a', wher e ao is nilpotent an d a' i s invertible. Applyin g thi s t o th e operato r a — A, where A is an eigenvalu e o f a, w e have X = X A (a)©X/ , where X{a) 7 ^ 0 and X' i s an a-invarian t subspac e i n which A is not a n eigenvalu e of a . Th e desire d resul t ca n no w b e obtaine d b y inductio n o n dim X applie d t o X'. ^ • Exercise. Verif y directly , using definition (3.1 5) , that th e subspaces X(a) ar e linearly independent . [Hint : Inductio n o n the lengt h o f the relation s xi- - xm = 0, where Xi G Xz(a) an dA ^7^ Xj whe n i 7 ^ j. Fo r a suitabl e n , appl y th e operato r (a — Ai)n t o suc h a relation an d the n appl y ( a — Ai)_n .]
- 35. 18 1. BASI C NOTION S Example. Th e eigenvector s o f a correspondin g t o differen t eigenvalue s ar e linearly independent . 3.8. Corollary . Each operator a G EndX can be uniquely written as (3.17) a = 6 + e, where S is diagonal, e is nilpotent, and 5 commutes with e : 5e = eS. PROOF. Settin g Sx — Xx whe n x G X(a) w e have a diagonal operato r S such that e — a — 5 is nilpotent (o n each X(a) an d henc e on X). Thi s implie s (3.1 7) . Conversely, fo r eac h decompositio n (3.1 7 ) (wit h th e mentione d properties ) X is a direc t su m o f th e subspace s X(5) = ker(5 — A) , which ar e a-invarian t sinc e aS — 5a. Th e operato r e — a — 5 is nilpotent o n X(5) onl y whe n A is the uniqu e eigenvalue o f a o n X(5). Therefor e X(a) = X(5) fo r eac h A G F, whenc e th e uniqueness o f (3.1 7) . • Exercises. 1 . Eac h elemen t o f EndX commutin g wit h a also commutes wit h its component s 5 and e. 2. det a = de t 6. 3. I f a i s invertible, the n u = a8~ l i s unipotent, i.e. , u — 1-f x, wher e x n — 0. 4. Th e decomposition a = du, where 5 is diagonal, u is unipotent, an d 5u = uS, is unique. 3.9. Jorda n blocks . A n operato r a G End X i s sai d t o b e cyclic i f X i s spanned b y the vectors ak xo (k G Z+) fo r som e xo G X (i n that cas e XQ i s also said to b e cyclic). Clearly , i f a i s cyclic , the n X i s spanne d b y th e vector s a k xo wit h 0 < k < n — 1 , where n = di mAT is the largest numbe r k such that th e vectors a k Xo (0 < k < n — 1 ) ar e linearly independent . If a is nilpotent, the n a n xo = 0 and th e matri x o f a in the basi s e% = a n ~l xo i s of the followin g form : (3.18) j n(0) This matri x i s a special cas e of a Jordan matrix (o r Jordan block) (3.19) jn(a ) = a + j n(0), which i s obtained fro m j n(0) b y th e substitutio n 0 ^^ a fo r th e diagona l element s in (3.1 8) . Exercises.1 . Verif y that , fo r a = j n(0), th e subspac e kera m (respectively , imam ) coincide s wit h th e linea r spa n o f the basi s vector s ei,... , em (respectively , e,..., e n _m j. 2. Verif y tha t th e operato r a = j n(0) i s indecomposable, i.e. , i t canno t b e written a s diag (ai, c^) fo ra ^7^ 0 (i — 1, 2). 3.10. Theorem . Any operator a G EndX is a direct sum of Jordan blocks O'i — jni{&i)> where on are the eigenvalues of a {the sum of allni equals n — dimX). /o 0 0 1 0 0 . . 0 1 0 . . 0 0 0 . 0 0 0 . . 0 . 0 .1 . 0)
- 36. 3. ELEMENT S O F LINEA R ALGEBR A 19 PROOF. B y Theore m 3.4 , i t suffice s t o assum e tha t a has a uniqu e eigenvalu e a. Replacin g a by a — a w e may als o assume tha t a — 0, i.e., a is nilpotent . First w e fix 0 7^ xi G X an d se t I = l i ® l 2 , wher e X i s the linea r spa n of ak x (k G Z+ ). The n a can b e writte n a s i n (3.9) , wher e a = j n i (0) an d (b y th e induction assumption ) 5 is the direc t su m o f / Jordan block s ( / G Z+). Assume tha t n — dimXi i s the larges t possibl e amon g th e dimension s o f th e similarly constructed summand s o f X. W e shall now show that equatio n (3.1 1 ) ca n be solved, an d therefor e a = diag (a , 5). Repeating th e argument s o f 3.4 , i t suffice s t o assum e tha t 1 = 1 . I n tha t case 5 = jn 2(0)? wher e n 2 < n. Therefore , X 2 i s spanned b y th e vector s a k x2-> wher e an2 x2 G X1. Not e tha t ani x2 = ani -n2 (an2 x2) = 0 (since ani = 0 on X). B y Example 1 of 3. 9 (o r by direct calculation) , we have tha t an2 x2 i s the linea r spa n o f ak x fo r k > n2. Therefore , an2 x2 = a n2 x0 for som e XQ G X. Replacin g x 2 b y x2 — XQ w e have XQ — 0. The n X 2 i s invariant , i.e., a — diag (a , 5). Now note that th e substitution x 2 ^ x 2 — XQ is of the form (3.1 0 ) an d therefor e equation (3.1 1 ) i s solvabl e fo r I = 1 . I n th e genera l cas e th e solutio n o f (3.1 1 ) i s of the for m x = (x, ..., £/) , wher e Xi is a solution o f (3.1 1 ) wit h S = ^ . W e now conclude tha t a — diag (ce , 5i,..., Si). D 3.11. Definition . A representatio n o f a G EndX a s a direc t su m o f Jorda n blocks jm{&i) i s called th e Jordan normal form o f a. Exercise. Verif y tha t th e numbe r o f block s i n th e Jorda n norma l for m o f a nilpotent operato r a equals di m (ke r a). 3.12. Projectio n operators . A n operato r p G EndX i s calle d a projection operator i f p 2 = p. I n thi s cas e p' = 1 — p i s als o a projectio n operator , an d p'p = pp' = 0. Fro m thi s i t follow s (exercise ) tha t X=pX®p'X, showing that p projects X ont o pX paralle l t o p'X. In general , an y decompositio n o f X int o a direc t su m o f subspace s X % (i G /) gives rise to projectio n operator s Pi(x) = x tl wher e Xi (i G /) i s the componen t o f x G X i n Xi. Therefore , (3.20) 1 = ^2Pi, PiPj = 0 whe n i ^ j . i With ou r blanke t assumptio n dim X < oo , we should assum e card / < oo . It i s easy t o chec k (exercise ) tha t th e su m s = p + q, where p,g G End X ar e projection operators , i s a projectio n operato r i f an d onl y i f p an d q are relate d b y the orthogonality relation: p _L q: pq = qp — 0. I n tha t cas e sX = pX 0 qX.
- 37. 20 1. BASI C NOTION S Similarly, th e differenc e r = p — q i s a projectio n operato r i f an d onl y i f p an d q ar e relate d b y th e order relation: q < p: pq = qp = g , whic h i s equivalen t t o pX = qX © rX, i.e. , qX C pX. 1 A famil y o f projection operator s (pi)iei i s said t o b e orthogonal i f the secon d condition of (3.20) is satisfied. A family (pi)iei i s called an orthogonal decomposition of identity i f both condition s (3.20 ) ar e satisfied. Clearly , in that cas e X i s a direc t sum o f the subspace s Xi = piX. Exercises.1 . A n operato r a G EndX i s diagonal , a = dia g («i,... , an ), i n the basi s e — (ei,..., en) i f and onl y if n (3.21) a = YlajPi, where Pi is the projectio n o f X ont o Xi = Fe{ (i = 1 ,... , n). 2. I f oti 7^ aj whe n i ^ j , the n th e operator s pi ca n b e written a s a — a* (3.22) Pl =n (%i — a n- 3. Sho w tha t eve n i n th e genera l case , th e projectio n operator s pi : X — >A Q in (3.8) are polynomials in the operator a. [Hint : Us e the operators pik — {a—i) k qi, where (3.23) Qi = [J(a, -aj) n '" with exponent s n ^ = dimX ^ ( 0 < k < rii — 1 ). ] 4. Functiona l calculu s 4.1. Th e algebr a F[x. Le t F[x ] b e a vector spac e ove r a field F wit h basi s the symbol s x k , wher e k G Z+. Element s / G F[x] ca n b e uniquel y writte n a s (4-1) /(* ) = ! >xf c , fc=0 with /fc G F, an d are called (formal) polynomials in the independent variable x = x The relatio n x k xr = x fc+r , wher e k,r £ Z + , uniquel y extend s (b y bilinearity ) to a multiplicatio n i n F[x] suc h tha t F[x] become s a n associativ e (commutative ) algebra wit h identit y x° — 1over F. Polynomials (4.1 ) admit substitution s x na (i.e. , xk -*ak ) fo r each fixed a G A, where A i s an associativ e algebr a ove r F. Mor e precisely , n (4-2) /(a ) = ]T/ fcafe . fc=0 The multiplicatio n rul e in F[x] shows that 7 : f(x) ^ f(a) i s an algebra homomor - phism fro m F[x] t o A, whic h i s uniquely determine d b y th e substitutio n x1—» a. In particular , fo r eac h A G F th e numbe r /(A ) i s called th e value of f G F[x] at X e F. Thu s th e ma p A> — >• /(A) , fo r a fixed / , i s a numerica l polynomia l (o r a polynomial functio n / : F — • F). 1 The las t tw o statement s ar e vali d onl y i f th e characteristi c o f th e bas e fiel d i s differen t from 2 . Editor's Note.
- 38. 4. FUNCTIONA L CALCULU S 21 If F is infinite, the n (a s is easily verified usin g the Vandermonde determinants ) the numerica l function s /^(A ) = k ar e linearly independent . I n that cas e the ma p f(x) i- > /(A) i s an isomorphism betwee n F[x] an d th e algebra P(F ) o f numerical polynomials / : F —> F. In general , th e map 7: f(x) 1— > f(a) (fo r a fixed a G A) is called a functional calculus (o f class F[x] ) i n A. Similarly , an y extension o f 7 to a homomorphis m 7: $—> A, where $ is an extension of F[x], is called a functional extension of class $ i n A. For eac h polynomia l / G F[x ] it s derivative f G F[x ] i s defined b y the rule [xk )' — kxk ~1 fo r each k G Z+, i.e., n (4.3) /'(* ) = Efc/fe^-1 . fc=0 Similarly one defines th e higher-order derivatives f^ — (f<<rn ~1 "> y, wher e m G Z+ and /(° ) = /. Not e that /( n + 1 ) = 0 for each polynomia l (4.1 ) . Later w e will also need th e followin g analo g of the Taylor formula i n F[x. 4.2. Proposition. Let cha,r F — 0 andx andy be two independent (commuting) variables. Then, for each f G F[x], (4.4) /( * +y) = £ ^ r V , A;= 0 where n is the degree of polynomial (4-1 )- PROOF. Sinc e bot h side s of (4.4) depen d linearl y o n /, i t suffices t o consider the cas e f(x) = xn . The n /(/c) (x) = n(n - 1 ) • • • (n - k + l)x n 'k for eac h k G Z+ , an d (4.4) become s Newton' s binomia l formul a (4.5) (z+y r = £(™y-v, where (^ ) are the binomial coefficients . Formul a (4.5 ) is proved by induction o n n. For thi s it is essential tha t x an d y commute. Th e assumptio n cha r F = 0 implies that k ^ 0 for each k G Z+ (i n particular, 0 ! = 1 ) . Thu s (4.4 ) i s proved fo r al l / G F[z] . • 4.3. Function s o f operators. Assum e (fo r th e sak e of simplicity) tha t F = C, an d let X b e a finite-dimensional vecto r spac e ove r C . Recal l (se e 3.8 ) that each a G EndX ha s canonica l decompositio n (3.1 7) , where Se = e6. Th e smalles t number m G Z+ such that £ m + 1 = 0 is called th e nilpotency height o f a = S +£. Let m = 0 , i.e., a = 6 = dia g (c*i,... , an ) i n some basi s e in X. Fo r each function / : F—> F w e set (4.6) /(a) x = /(A)x whe n x G XA(a), i.e., /(a) = diag (/(ai),... , f(a n)) i n some basis e. Thi s definition doe s not depen d on the choic e of the basi s e in X.
- 39. 22 1. BASI C NOTION S In th e genera l case , le t <& m b e th e algebr a o f function s / : F — > F havin g derivatives f( k &i) u p t o orde r m a t th e point s o ^ (i — 1,... , n). Her e m i s th e nilpotency heigh t o f a. Fo r / G ^m w e set (4.7) f{a) = £ /( / c ) W fc! fc=0 -£ it Here th e f^(S) ar e define d a s i n (4.6 ) (replacin g / b y f ^ ) . Applyin g Proposi - tion 4.2, we find that (4.7 ) holds for al l polynomials / G F[x. Thu s definition (4.7 ) is an extensio n o f definition (4.2 ) t o function s o f class<3>m . Examples. 1 . Fo r a Jordan bloc k a = jn(a) definitio n (4.7 ) become s (4.8) /(a ) = £^/( f c ) (« ) . ife! fc=0 e where e = j n(0). Clearly , /(a) i s an upper-triangular matri x with entries f^ k a)/k in the kth stri p paralle l t o the mai n diagonal . Fo r example , wit h n — 3 / / ( a ) /'(a ) |/"(a) > (4.9) /(a ) 0 /(a ) /'(<* ) 0 0 f(a) 2. Le t /(A ) = e A . Usin g (4.8 ) an d th e Jorda n norma l for m o f a we have (4.10) e a = es ee , wher e e £ = ^ ^ . fc=0 Exercise. Prov e (usin g Example 3 of 3.1 2 ) tha t 6 and e are polynomials i n a. 4.4. Theorem .XTi e map 7 : / H- > /(a) ^we n6? / (4.7) z s a functional calculus of class &m, where m is the nilpotency height of a G EndX. PROOF. I t suffice s t o check the multiplicative property o f 7: <£ m—> EndX. T o this en d w e utilize th e Leibni z rul e (4.11) (fg) W = £ ( * ) / W $ W for /, # G $m . Recal l tha t (£ ) = 751 - I n particular , (J ) / 0 only whe n 0 < s < k. Replacing / b y fg i n (4.7) , we have (4.12) (/ g)(a) = SE f{S) ^f{S) ^ = f^a), k s+t=k i.e., ^y(fg) — j(f) • l{g)> Therefore , 7 i s a n algebr a homomorphis m fro m<£> m to EndX. D Example. Settin g ft(X) = e~ A , wher e t G F = R,C , w e obtai n a famil y o f operators w(t ) = e ta ( a G EndX) satisfyin g (4.13) u(t + s) = u{t)u(s), u(Q) = l for al l t,s e F. Exercises.1 . Verif y tha t th e functio n u(t) — eta i s differentiabl e i n t G F (holomorphic i f F = C ) an d fo r al l t G F, (4.14) u'(t ) = au(£ ) =u(t)a .
- 40. 4. FUNCTIONA L CALCULU S 23 2. Le t Nil X (respectively , UniX ) b e th e se t o f al l nilpoten t (respectively , unipotent) operator s a G EndX. Prov e tha t th e formula s k — (-l) k xk (4.15)e X = £fcT H 1 + *) = E where x G NilX, defin e bijection s betwee n Nil X an d UniX . We als o poin t t o th e mor e genera l multiplicativ e propert y fo r operato r expo - nents i n EndX . 4.5. Proposition . If the operators a , b G EndX commute, then (4.16) e a eb = e a +b . PROOF. I f a an d 6 are nilpotent , the n (4.1 6 ) ca n b e checke d directl y usin g finite sum s (4.1 0) . I n th e genera l case , notic e tha t eac h roo t subspac e X(a) i s 6-invariant (sinc e ab = ba). Therefor e (4.17) I = 0 1 v , where X^ i s the intersectio n o f X(a) an d X^{b). Moreover , eac h X^ M i s both a- and 6-invariant . Restrictin g thes e operator s t o X^ w e hav e tha t a = A + e an d b = jjb + e', wher e e and e' commute . Therefor e ea eb = e x+ ^e£ e£ ' = e x +^es +£ ' = e a +b '. • Remark. Definitio n (4.7 ) becomes more natural i f <£m is replaced b y the alge- bra $ o f formal powe r series (4.1 ) with n = oo . On e can show that th e convergenc e of th e serie s f^ k ) a t point s A = ce^ , where 0 < k < m , implie s (component - wise) convergenc e o f operator serie s (4.2) . Fo r example , th e operato r exponen t e a (a G EndX) i s given b y the convergen t serie s oo n (4.18) e a = 2 We shal l revisi t thi s proble m i n 1 3.8 . I n th e meantime , w e shal l mentio n ye t another approac h to the functional calculu s in EndX, whic h works even in the cas e dimX = oo . 4.6. Norme d space . Le t X b e a vecto r spac e ove r a field F = R , C. A numerical function p: X— > R is called a seminorm o n X i f the following condition s are satisfied : (a) p(Xx) = Xp(x) for al l A G F, x G X; (/?) p( # + y ) < p(x) + p(v) fo r al l x, y G X . If A = 0 , the n (a ) show s tha t p(0 ) = 0 . I f x + y = 0 , the n (/3 ) show s tha t p(x) > 0 for al l x G X, i.e. , p takes o n values i n R+ = [0 , +oo). A seminorm p i n X i s called a nor m i f p(rr) = 0 only when x — 0. I n tha t cas e one uses the notatio n p{x) = x (similar t o |A | in F). I t i s easy t o chec k tha t (4.19) p(x,y) = x-y is a metri c o n X . Accordingly , ||x| | = p(x,0) i s interpreted a s the length o f x (th e distance fro m x t o 0 G X).
- 41. 24 1. BASI C NOTION S X i s called a normed space if a norm p(x) = ||x| | i s defined o n it. Accordingly , metric (4.1 9 ) i s given on X. A normed spac e X i s a Banach space if X i s complete with respec t t o metri c (4.1 9) . Exercises.1 . Prov e tha t th e vecto r operation s o n a norme d spac e X ar e continuous (wit h respec t t o metri c (4.1 9)) . 2. Prov e tha t eac h seminor m p on X satisfie s (4.20) p(x) - p(y) < P(x - y) for al l x,y G X. 3. Prov e (usin g (4.20) ) tha t th e functio n p(x) = x is continuou s (o n th e normed spac e X). Examples. 1 . Th e spac e X — Fn i s Banach wit h respec t t o each nor m ^ i / p (4-21) INI P = { E W P ) 2 = 1 where 1 < p < oo . 2. Le t lp be the set of all sequences x — (#1 ,... , xi,...), wit h X{ £ -F , satisfying the conditio n oo (4.22) J2^ P <OG - i=l The spac e X = Z p is Banach wit h respec t t o nor m (4.21 ) , assuming n = oo . 3. Le t C(a , 6) be the space of all continuous (F- valued) function s o n [a , b]. The formula (4.23) H/l l = ma x |/(i) | a<t<b defines th e uniform norm o n C(a,b), wit h respec t t o whic h C(a,b) i s a Banac h space. 4. Le t T b e a se t o f Lebesgu e measur e /i , an d L p(T,n) th e vecto r spac e o f measurable function s / : T — > F suc h that |/| p i s integrable. The n th e formul a (4-24) H/llp={/l/(t)l P dM(<)} " defines a seminorm o n L v(T,ji). I t take s valu e zer o on th e function s / ~ 0 , wher e f ~ g means tha t / an d g are equal almos t everywher e (wit h respec t t o //) . Semi - norm (4.24 ) become s a nor m i f the function s / ~ 0 are considere d t o b e zero . I n that sens e L P(T,IJ) i s a Banach space . The proofs o f these assertions can be found i n textbooks o n functional analysi s (see, fo r example , [43] , [66]) . Norm s (4.21 ) an d (4.24 ) ca n b e extende d t o includ e the valu e p — oo. Fo r example, i n case (4.21 ) w e set (4.25) Hallo o =supjx,j . i Similarly, Loo(T , /j,) is define d a s th e spac e o f essentiall y bounde d (i.e. , bounde d almost everywhere ) measurabl e function s / : T — > F. The n Loo(T,/i ) i s a Banac h space with respec t t o th e nor m (4.26) 1 / 1 1 00 = in f (su p |/(t)|), T~S S where T ~ S mean s fi(T S) = 0 .
- 42. 4. FUNCTIONA L CALCULU S 25 In particular, le t T — [a, b] an d let /i be the standard Lebesgu e measure (gener - ated b y the Euclidean measure on [a , b). I n that case , LP(T, fi) i s denoted L p(a, b). As another notatio n w e mention L P(R). Exercises.1 . Verif y (usin g th e inequalit y a + b < 2max(a , b) fo r a — xi, 6 = yi) tha tZ p (similarly, L v(T,n)) i s a vector space . 2. Verif y tha t th e convergenc e f n— » / i n C(a , 6) i s th e unifor m convergenc e /«(*) = 4 f{t). 3. Verif y tha t C(a, b) i s everywhere dens e in L p(a,b) ( 1 < p < oo). 4.7. Th e algebr a B(X) . Le t X b e a norme d space . Fo r eac h a G L(X) = End X w e set (4.27) ||a| | = su p-^—r^- = su p ||ax|| . x^O X || x|| = i An operato r a G EndX i s said t o b e bounded i f ||a| | < oo , i.e. , ||ax| | < Cx fo r each x G X (wher e C > 0 is a constant) . It i s eas y t o chec k tha t th e boundednes s o f a G L(X) i s equivalen t t o th e continuity o f a o n X . Th e spac e B(X) o f al l bounde d operator s a G L(X) i s a normed spac e wit h respec t t o nor m (4.27) . Furthermore , B(X) i s an algebr a ove r F suc h tha t (4.28) || a&|| < ||a| | • ||b|| for al l a,b G B(X). Repeatedl y applyin g (4.28 ) w e have (4.29) ||a n || < a n for eac h n G Z+ . Not e tha t ||1 | | = 1 . If X i s complete, the n B(X) i s also complete (see , for example , [43] , [66]) . I n that case , i t i s easy t o check , usin g (4.29) , that fo r eac h serie s (4.1 ) (wit h n = oo ) operator serie s (4.2 ) converge s i n B(X) whe n ||a| | < r(/) , wher e r(f) i s the radiu s of convergence o f (4.1 ) . Let a G B(X) an d le t $ a b e th e algebr a o f powe r serie s (4.1 ) wit h radiu s o f convergence r(f) > a. I t i s eas y t o chec k tha t th e ma p 7 : f(x) -* f(a) i s a functional calculu s of class $ a . Examples.1 . Th e function s /(A ) = ( 1 — A)- 1 expand s int o a geometri c progression wit h radiu s o f convergence 1 . Therefore , th e operato r 00 (4.30) /(a ) = £ V n=0 is define d wheneve r ||a| | < 1 . A direc t calculatio n show s (exercise ) tha t f(a) — ( l - a ) " 1 . 2. Th e exponentia l /(A ) = e x i s given b y a power serie s with radiu s o f conver - gence oo . Therefore , th e operato r exponentia l 00 n (4.31) f(a) -e^ = J2~ n=0 is defined fo r eac h a G B(X).
- 43. 26 1. BASI C NOTION S 4.8. Th e cas e dim X < oo . Seminorm s p and q on a vector spac e X ar e sai d to b e equivalent i f (4.32) p(x) < Aq(x), q(x) < Bp(x) for each x Gl (fo r som e constants A,B>0). I t i s easy to check (see , for example , [72]) that , whe n dim X < oo , all norms o n X ar e equivalent . Using inequalities (4.32 ) for the norms p and q on X w e find that p-convergenc e Jbfi ' Jb l o equivalen t t o ^-convergenc e x n— » x. Therefore , t o verif y convergenc e in X (dim X < oo ) i t suffice s t o fix one of the norm s (4.21 ) o r (4.25) . Therefore, B(X ) = L(X ) fo r eac h nor m o n X (i.e. , al l linea r operator s ar e continuous whe n dim X < oo) . Consequently , w e can conside r functiona l calculu s of class <£ a (se e 4.7) whe n a G L(X). For example , exponentia l (4.31 ) i s defined fo r eac h a G L(X). Exercises. 1 . Suppos e dim X < oo . Verif y tha t definitio n (4.2 ) wit h / G &a (n = oo) coincide s wit h definitio n (4.7) . 2. Verif y (usin g (3.4) ) tha t (4.33) det(e a ) = e t r a , where tra (th e trace of a) i s defined (simila r t o 3.1 ) a s trae fo r eac h basis e of X. 3. Verif y (usin g the Jorda n norma l for m o f a) tha t (4.34) e a = lim( l + - ) n . Henceforth we shall also use the notation B(X, Y) fo r the vector space of all bounded (i.e., continuous) operator s a G Horn (X, Y). 5. Unitar y space s 5.1. Definition . Le t X b e a vecto r spac e ove r a field F = R,C . A functio n / : X x X —> F is called a Hermitian form o n X i f for all a, (3 G F and all x,y,z G X, (a) f(x,y) = f(y,x), an d (/?) /(a x + 02/, z) = a/(:r , *) + /?/(y , z). Here th e overba r denote s comple x conjugatio n i n F. Condition s (a) an d (/3 ) imply tha t / i s antilinear i n the secon d argument , i.e. , /(x, ay + pz) = a/(x , y) + fif(x, z). A Hermitian for m / i s called a scalar product o n X i f it i s positive definite , i.e. , (7) /(# , #) > 0 and f(x, x) — 0 implies x = 0 . The space X i s said to be unitary i f a scalar product / i s defined o n it. I n tha t case, on e ofte n use s th e simplifie d notatio n f(x,y) — (x,y). Conditio n (7 ) allow s us to defin e a nonnegative functio n (5.1) ||x| | = V ^ ) - If F — M, then th e comple x conjugatio n i n (a ) an d (/? ) i s trivial , an d / i s a symmetric bilinea r for m o n X . A unitar y spac e ove r R i s als o calle d a Euclidean space. 5.2. Proposition . Function (5.1 ) is a norm on X [so that X is a normed space over F). Furthermore,
- 44. 5. UNITAR Y SPACE S 27 (a) For any two vectors x,y G X, the Schwarz inequality holds: (5.2) l(*,J/) l < IM I • IMI- (/?) Equality occurs in (5.2 ) only when x and y are collinear. Moreover, for each x G X, (5.3) ||s| | =max ' 7 ^ = max|(x,z)| . v±o y Nl= i PROOF. Th e nonnegativ e functio n x — y2 coincide s with th e quadrati c tri - nomial ^(A) = H 2 - 2 R e ( A - ( * , j / ) ) + |A| 2 ||y||2 , where A G C. Th e substitutio n A = te ia , wher e a i s determine d b y (x,y) = pe za , makes (p(X) into a nonnegative trinomia l Mt) = x 2 -2tp + t 2 y2 , with t G l , whos e discriminan t A mus t b e nonpositive , whic h give s (5.2) . Thi s proves (a). Furthermore, the equality A = 0 implies that (p(X) = 0 for someA, i.e., x = Ay , whence (/?) . I t i s clear that "triangl e inequality" (/? ) of 4.6 for function (5.1 ) follow s from (5.3) . I t i s als o clea r tha t conditio n (a ) o f 4. 6 holds . Therefore , (5.1 ) i s a norm o n X. • Examples. 1 . Th e spac e F n i s unitary wit h respec t t o th e scala r produc t n (5.4) {x,y) = ^2xiyi. 2 = 1 2. Th e spac e I2 (see 4.7 ) i s unitar y wit h respec t t o scala r produc t (5.4) , as - suming n = 00 . I n tha t case , (5.4 ) converge s absolutely , whic h follow s fro m th e inequality (5.5) ab< -(a 2 + 6 2 ), where a = |x^| , b = |^| . 3. Th e spac e I/2(T , //) i s unitary wit h respec t t o th e scala r produc t (5.6) (/,<?) = J fit)g(t)d^t). Again, on e has t o us e (5.5) , this time wit h a = f(t) an d b = g(t). Note that th e Schwarz inequality in I2 and Z/2(T, /i) coincides with the Cauchy- Bunyakovsky inequality. Exercises.1 . Verif y (usin g th e Schwar z inequality ) tha t th e scala r produc t on a unitary spac e i s continuous (a s a function o f two variables) . 2. Nor m (5.1 ) satisfie s th e parallelogram rule (5.7) || x + y|| 2 + ||a ;-2/||2 = 2(||x|| 2 + || y||2 ) (for al l x,y G X). Moreover , th e polarization rule (5.8) Re(x,y)= 1 l(x + yf-x-y 2 ) holds fo r al l x, y G X.
- 45. 28 1. BASI C NOTION S 3. A norme d spac e X i s unitarizable (i.e. , the nor m i s of the for m (5.1 ) ) onl y when (5.7 ) holds . 4. C(a, b) is not unitarizable . 5. Th e spac e l p (and , similarly , L p(T,p)) i s unitarizable onl y whe n p = 2 . 5.3. Th e spac e faiw). Ye t anothe r exampl e o f a unitarizabl e spac e ca n b e obtained i f the countabl e serie s (i n the definitio n o f I2) are replace d b y (5.9) a = y^Qj , iei where ai G F an d I i s an arbitrar y se t o f indices. More precisely, a partial sum o f series (5.9 ) is a number ap obtaine d fro m (5.9 ) by replacing I wit h a n arbitrar y subse t F C I. I fa ^ > 0 (for eac h i G J), the n th e sum o f (5.9 ) i s defined a s a — su p Fa p.Accordingly , w e say tha t serie s (5.9 ) con- verges if a < co. I n th e genera l case , we may similarl y defin e absolute convergence for (5.9) . Setting uo — card/, le t l p(u) ( 1 < p < co ) b e th e se t o f al l x = (xi) iei, wit h Xi G F. Assum e tha t th e serie s compose d o f th e number s xi p converges . The n one easil y verifie s (simila r t o l p) tha t l p(u>) i s a Banac h spac e wit h respec t t o th e norm (5.10) M P = { E W P } 1 / P - iei In particular , faiw) i s unitar y wit h respec t t o scala r produc t (5.4) , wher e th e finite su m i s replaced b y the absolutel y convergen t serie s composed o f the element s Exercise. I f serie s (5.9 ) converge s absolutely , the n i t ha s a t mos t countabl y many term s whic h ar e differen t fro m zero . In particular, fo r eac h x G IP(UJ) at mos t countabl y man y coordinates Xi (i G /) are nonzero . 5.4. Orthogona l systems . Le t X b e a unitary space. Tw o elements x,y G X are sai d t o b e orthogonal (x _ L y)i f (x,y) = 0 . A system o f element s e = ( e ^ 6 / in X i s sai d t o b e orthogonal (respectively , orthonormal) i fe ^_ L Cj for i ^ j (respectively, (e^e^ ) = Sij). Let e be a n orthonorma l syste m i n X. Fo r eac h x G l, th e number s (5.11) Xi = (x,ei) are calle d th e coordinates (o r Fourier coefficients) o f x (relativ e t o e) . I t i s no t difficult t o sho w that, fo r eac h x G X, th e Bess el inequality (5.12) Tx z2 <xf holds. Indeed, th e definitio n o f th e su m o f serie s (5.9 ) show s tha t i t i s sufficien t t o prove (5.12) in the case card / < co . I n that case , we set i" = {1 ,... , n) an d consider the vecto r n
- 46. 5. UNITAR Y SPACE S 29 such that yi = xi (i = 1 ,... , n). Th e vector z — x — y is orthogonal t o the syste m e [z % = 0 fo r eac h i — 1,... ,n). Therefore , x = y + z, wher e y _ L z. Clearly , ||x||2 = ||y|| 2 + ||^|| 2 (th e Pythagorean Theorem). Als o y 2 coincide s wit h th e left-hand sid e of (5.12). Whenc e (5.1 2 ) (a s y 2 < ||x|| 2 ). Applying (5.1 2) , we have that fo r each x G X at most countably many coordi- nates Xi (i G I) are different from zero. An orthonormal system e in X i s said to be complete (respectively , closed) if its linear spa n i s everywhere dens e i n X (respectively , i f Zi = 0 for each i G / implie s that z — 0). Th e system e is called a n (orthonormal) basis of X i f each x Gl ca n be written a s a series (5.13) x = y^Cjej , where at mos t countabl y man y coefficient s Q G F are different fro m zero . Using the continuity o f the scalar produc t o n X w e have that Q = Xi for eac h i £ I (i.e. , the coefficients i n (5.13) are uniquely determined) . Similarly , one checks that (5.14) H 2 = 5>,|2 . iei Examples. 1 . Lete^ G h(^) b e the vector wit h coordinate s 5ij (j G I). The n the syste m e — (e^)^/ i s an orthonormal basi s in hi^u). 2. Th e function s (5.15) e n{t) = -^=e M , V Z7 T where n G Z, form a n orthonormal syste m i n 1/2(0, 27r). We shal l no w show tha t (5.1 5 ) i s a complete system . Indeed , th e linea r spa n T(0,27r) o f (5.1 5 ) i s th e algebr a o f trigonometri c polynomial s i n L 2(0,27r). Ac - cording to the Weierstrass Theorem , T(0 , 2n) i s everywhere dens e in (7(0, 2TT) with respect t o th e unifor m metric . Bu t the n i t i s also everywher e dens e wit h respec t to th e metri c i n 1 ^(0 , 27r). O n th e othe r hand , C(0,27r ) i s everywher e dens e i n 1/2(0, 2TT) (se e 4.6). W e now conclude that T(0, 2TT) is everywhere dense in 1/2(0, 2ir). 5.5. Hilber t spaces . A unitary spac e H i s called a Hilbert space if it is com- plete (wit h respec t t o norm (5.1 )) . In particular , al l unitary space s considere d i n 5.1 and 5.2 are Hilbert spaces . Unitary space s X, Y ove r a field F ar e said t o be isomorphic i f there exist s a linear isomorphis m u: X — • Y preservin g th e scalar product , i.e. , (5.16) [ux.uy) = (x,y ) for al l x,y G X. Exercises.1 . Conditio n (5.1 6 ) i s equivalent t o u bein g a n isometry, i.e. , t o the conditio n ux = ||x| | for each x G X. [Hint : Us e the polarization rule. ] 2. I f a unitary spac e X i s isomorphic to a Hilbert spac e Y, then X i s a Hilbert space. 5.6. Lemma . Let H be a Hilbert space, e — (e^)^/ an orthonormal system in H, and Xi (i G I) a system of numbers such that series (5.1 2 ) converges. Then
- 47. 30 1. BASIC NOTION S there exists a vector (5.17) x = J2 x%Ci iei in H such that x % (i G /) are the coordinates of x (with respect to e). In that case, Bessel inequality (5.1 2 ) becomes equality (5.1 4) . PROOF. Recal l (se e 5.3 ) tha t a t mos t countabl y man y number s X{ (i G /) ar e nonzero. Therefore , i t suffice s t o conside r countabl e serie s of the for m oo (5.18) x = 2_. x %e i- 2 = 1 Let s n (n G N) b e a partial su m o f series (5.1 8) . B y the Pythagorea n rule , we have that s n — 5m||2 equals the corresponding remainder o f series (5.1 2) . I t no w follow s (as H i s complete) tha t serie s (5.1 8 ) converges . The othe r assertion s o f this lemm a were proved i n 5.4 . • Now we can stat e an d prov e th e mai n theore m o f Hilbert spac e theory . 5.7. Theorem . Each Hilbert space H has an orthonormal basis. Moreover, if e is an orthonormal system in H, then the following are equivalent: (i) e is complete; (ii) e is closed] (iii) e is a basis of H. PROOF. Th e orthogonalit y relatio n z A- e (i.e. , Zi — 0 fo r eac h i G I) i s equivalent t o z 1 H{e), wher e H{e) i s th e closur e o f th e linea r spa n o f e. I f (i ) holds, the n z _ L H. I n particular, z _ L z, i.e. , z — 0 and (i )= > (ii). Th e implicatio n (ii)= > (iii) follows from Lemm a 5.6 . Indeed , fo r each x G H le t y G H b e the vecto r defined b y (5.1 7) , wher e X{ are th e Fourie r coefficient s o f x. Then , fo r z — x— y, we have tha t z _ L e, i.e. , z = 0 . Thu s x = y i n (5.1 7) . Th e implicatio n (iii )= > (i) follows directl y fro m th e equalit y x — limn sn , wher e th e s n ar e th e partia l sum s of (5.1 8) . Thu s assertion s (i) , (ii) , and (iii ) ar e equivalent . Using Zorn' s lemma, we see that ther e is a maximal (wit h respect t o inclusion ) orthonormal syste m e m H. I f 0 ^ z ± e , the n th e uni t vecto r z 0 — ^/IMI extend s the syste m e , which i s impossible. Therefore , e is a basis of H. • Example. Syste m (5.1 5 ) i s an orthonormal basi s in L2(0, 2TT). I n other words , each functio n / G I/2(0, 2TT) can b e written a s a Fourier serie s oo (5.19) /(*) = Yl c nemt , which converge s i n the metri c o f L2(0, 27r), with coefficient s (5.20) c n = ~J f(t)e- tnt dt. Moreover, equalit y (5.1 4 ) become s th e Plancherel formula (5.21) _ / f(t) 2 dt= ] T c n •'O <n.= — nn 5.8. Corollary . Each Hilbert space is isomorphic to /2(^) {for a suitable u).
- 48. 5. UNITAR Y SPACE S 31 Let e b e a n orthonorma l basi s o f H. B y Lemm a 5.6 , th e ma p ux = (xi)i ei (with respec t t o e) i s an isometr y betwee n H an d h^)- B y Exercis e 1 in 5. 5 (o r by direc t calculation) , w e have H ~ h(^)- It i s no t difficul t t o chec k (simila r t o 2.2 ) tha t an y tw o orthonorma l base s o f H hav e the same cardinality. Therefore , th e isomorphis m H « h{u) give s rise to a unique cardina l numbe r to — dim if, calle d th e Hilbert dimension o f H. In general , th e algebrai c dimensio n o f H (se e 2.2) ma y diffe r fro m th e Hilber t dimension dimH. However , i f one of them i s finite, the n the y coincide . Examples.1 . Th e Hilber t spac e L 2(a,b) (a ^ b) is of countabl e dimension . This i s clear fro m th e consideratio n o f Fourier serie s (5.20) . 2. A spac e H i s sai d t o b e separable i f i t contain s a countabl e everywher e dense subset . The n H contain s a finite o r countabl e linearl y independen t syste m fi (i G I). Applyin g th e Gram-Schmid t proces s w e see that H ha s eithe r a finite or a countable orthonorma l basis . We conclude that eac h separable Hilber t spac e is isomorphic t o either F n (n G N) o r l 2 (ove r F). Exercises. 1 . Verif y tha t L 2(a,b) an d Z/2(M ) are separable . 2. Le t ei (i G I) b e a n orthonorma l basi s i n H. Wit h eac h operato r a G B(H ) we associate a numerical matri x (5.22) a,ij = (ae j:ei). Verify tha t th e actio n y = ax o n H i s expressed i n terms o f the Fourie r coefficient s via the standar d rul e (5.23) yi = ^dijXj, 3 where the serie s i n the right-han d sid e converges absolutely . 5.9. Orthogona l sums . Le t Hi (i G / ) b e a n arbitrar y famil y o f Hilber t spaces ove r a field F. Le t (5.24) # = 0 # i iei denote th e set o f all sequences x — (xi)iei, wher e Xi G Hi an d th e serie s compose d of the number s ||a^|| 2 converges . I t i s easy t o chec k (simila r t o h{^)) tha t H i s a Hilbert spac e with respec t t o the scala r produc t (5.25) (x,y) = ^2(xi, yi) iei (this i s an absolutel y convergen t series) . We cautio n th e reade r no t t o confus e (5.24 ) wit h th e direc t algebrai c su m o f the space s Hi (se e 2.9) . Eac h Hi obviousl y embed s i n H, s o that Hi _ J _ Hj whe n i ^ j . Th e spac e H i s calle d th e direct orthogonal sum o f th e Hilber t space s Hi (iei). Note tha t u = di m H i s th e su m o f th e cardinal s uji = di m Hi (i G /). Thi s gives us a general method o f representing a n arbitrary Hilber t spac e H a s in (5.24). We can just fix an orthonormal basis e of H an d decompos e i t into a disjoint unio n of th e subset se ^ (i G /). I n tha t case , Hi i s th e closur e o f th e linea r spa n o f th e subsysteme ^ (i G /).
- 49. 32 . BASI C NOTION S Example. Le t Ho C H b e a close d subspac e an d Hi = H$ it s orthogona l complement (i.e. , the se t o f all y G H suc h tha t y J_ HQ). The n (5.26) H = H 0@H 1 . A projectio n operato r p o n H i s calle d a n orthoprojector i f (px,y) = (x,py) for al l #,2 / G iif . Fo r suc h a projector , (px,p'y) = 0 fo r al l x,y G H, wher e pi — 1 — p (th e complementar y orthoprojector) . Consequently , w e have orthogona l decomposition (5.26) , where HQ = pH an d Hi — p'H. A famil y o f orthoprojector s (pi) %^i o n H i s sai d t o b e orthogonal i f p^j = 0 when i ^ j . I t i s easy t o chec k that , i n that case , the forma l serie s i converges a t eac h x G H. Moreover , p project s H ont o th e direc t orthogona l su m of the Hi—piH. I n particular , p — 1(the identit y operator ) i n (5.24) . Exercises.1 . I f p ^ 0 i s a n orthoprojector , the n ||p| | = 1 (i n particular , peB(if)). 2. Eac h nondecreasin g sequenc e o f orthoprojector s p n G B(H) converge s a t x G H t o a n orthoprojecto r p G B(iJ) . [Hint : Th e famil y Ap n = p n — pn_i (po = 0 ) i s orthogonal. ] 5.10. Theorem . Any continuous linear functional f G H' is of the form f(x) = (x,y), where y e H. Moreover, f — ||y||. The map y ^ f is an antilinear isomorphism H ^ H f . PROOF. Le t H 0 = ker/ . The n codimi7 0 = 1 if / ^ 0 . Thi s gives rise to (5.26 ) with one-dimensiona l Hi = FCQ, wher e 0 ^ e oJ_ #o- Settin g x — xo + Aeo , where xo G HQ and A G F, w e have f(x) = A/(e0), (x , e0) = A(e 0, e0), which implie s (5.26 ) wit h y = /(eo)eo an d ||eo| | = 1 . T o find the nor m o f / w e can use (5.3 ) (replacin g x b y y). Thi s give s ||/| | — y. D Exercise. Verif y tha t th e antilinea r isomorphis m H « H' abov e i s given b y (5.27) (x,y) = {x,y), where x i- » x i s th e comple x conjugatio n i n H (wit h respec t t o a n orthonorma l basis i n H). 5.11. Theorem . For each a G B(H) (see 4.7 ) t/ier e exist s a unique (adjoint) operator a * G B(if), defined by (5.28) (ax,2/ ) = (a:,a*y ) /or a/ / x,y E H. The map a -^ a* is antilinear and satisfies the relations (ab)* — 6*a* and a** = a . Moreover, a = ||a*|| . PROOF. Not e tha t f(x) = (ax,y) i s a continuou s linea r functiona l o n H, an d therefore /(# ) = (x ,2;) fo r som e (uniquel y determined ) z E H. I t i s clea r tha t z i s a linea r functio n o f y. Settin g z = a*y, w e hav e (5.28) . Recal l als o tha t
- 50. 5. UNITAR Y SPACE S 33 ||a|| = su p ||ax||, wher e th e supremu m i s take n ove r th e uni t spher e x = 1 . Therefore, ||a|| =sup|(ax,2/)| = su p |(x,a*y)| = ||a*|| , where the supremum i s taken ove r x = ||y| | = 1 . Th e other propertie s of the ma p a*-* a* follow easil y fro m (5.28) . • Exercise. A functio n / : H x H— » F i s said t o b e conditionally bilinear (o r sesquilinear) i f /(#, y) i s linear in x and antilinear i n y. Prov e that eac h continuou s conditionally bilinea r functio n / : H x H — > F i s o f th e for m (5.28 ) (fo r som e aeB(H)). 5.12. Definition . A n operato r a G B(if) i s said t o b e (respectively ) Hermit- iarij anti-Hermitian, unitary i f a* = a, a* = —a , a* = a - 1 (i n the latte r case , a is assumed t o b e invertibl e i n 3(H)). It is easy to check (exercise) that a * = a if and only if the form f(x,y) = (ax , y) is Hermitian and also if and only if the quadratic form <p(x) = (ax , x) is real-valued. Moreover, a * = a i f an d onl y i f matri x (5.22 ) i s Hermitian (fo r eac h orthonorma l basis o f H). The substitution ai —> ia (whe n F — C) yields a bijection between real subspaces of Hermitia n an d anti-Hermitia n operator s i n B(i7) . Fo r eac h a G B(jFf) , th e operators a i = o (a + a *)> a 2 = —( a - a* ) are Hermitian , an d w e hav e a uniqu e decompositio n a = a + ia,2- Th e operato r a G B(jff) i s unitary mean s that a is an automorphis m o f the Hilber t spac e H. Exercises.1 . I f a = a* , the n eac h eigenvalu e a o f A i s rea l an d an y tw o eigenvectors wit h differen t eigenvalue s a ^ (3 are orthogonal . 2. IfaHo C HQ (where HQ is a subspace of H), the n a*H$ C H$ . In particular, decomposition (5.25 ) i s invariant i f a — a*. 3. I f H i s a finite-dimensional unitar y spac e ove r C , the n eac h Hermitia n operator a G L(H) i s diagonalizable. [Hint : Us e Exercises 1 and 2. ] 4. An y Hermitia n operato r a G B(H) i s uniquely determine d b y it s quadrati c form (f a(x) = (ax,x) . I n othe r words , a = 6 < ^> ipa = ^ . [Hint : Expres s f a(x,y) = (ax,y) vi a</? a using (5.8). ] Example. A n operato r a G B(if) i s said t o b e positive (a > 0) i f (ax,x ) > 0 for eac h x G fl". I n particular , a = a* , i.e. , a i s Hermitian . I f d i m # < oc , the n a — diag (ai,... , an ), wher e c ^ > 0 ( z = 1 ,.. . ,n). 5.13. Compac t operators . A linear operator a G L(i7) is said to be compact if i t send s ever y bounde d se t A C H t o a precompac t subse t o f H. Equivalently , if Si i s the uni t bal l in H (i.e. , the set o f all x G H suc h that ||x| | < 1 ) , then aS i s precompact. Here w e us e th e followin g definitio n o f precompactnes s i n a metri c spac e H: a subse t N C H is precompac t i f it s closur e N i s compact . I n particular , eac h precompact set N C H i s bounde d (wit h respec t t o th e nor m o n H). W e no w conclude that an y compac t operato r a G L(iJ) i s bounded, i.e. , a G B(H). Example. Le t H = Z/2(T, /i), where /x(T ) < oc . A n integra l operato r (5.29) (af)(t)= [ K(t,s)x(s)dn(s),
- 51. 34 . BASI C NOTION S with K G L2(T2 ,/i2 ), i s called a Fredholm operator o n H. It i s eas y t o chec k (see , fo r example , [66] ) tha t Fredhol m operato r (5.29 ) i s compact. Moreover , th e Fubin i theore m implie s tha t a ha s a n adjoin t Fredhol m operator a* , generate d b y the functio n K*(t,s) = K(s,t). A s a direct consequenc e of (5.29) , we have a n estimat e (5.30) ||a| | < HATH, where ||i^| | i s the norm of K i n L2(T2 , /i2 ). Finally , the Fubini theorem als o implies that, fo r eac h orthonorma l basi s e n (n G N) o f H, th e matri x element s a mn — (aen, e m) o f a ar e given b y where e mn(£, s) — ern(t)en(s). Th e parenthese s i n th e right-han d sid e o f (5.31 ) stand fo r th e scala r produc t i n L 2(T2 ,/x2 ). Henceforth w e assum e tha t F = C . A basi s e = (ei)i ei o f H i s calle d a n eigenbasis wit h respec t t o a £ B(H ) i f aei = a ^ fo r al l i G /. 5.14. Theore m (Hilbert) . Any compact Hermitian operator a G B(i7) has an orthonormal eigenbasis in H. In other words, (5.32) # = 0 #A A (direct orthogonal sum), where H = ker(a — A). Furthermore: (i) If X ^ 0, then dimH < oc (i.e., the eigenvalue X has finite multiplicity). (ii) The number of different eigenvalues of a is either finite or countable. (iii) / / the number of eigenvalues of a is countable, then they form a sequence such that X n— > 0. For a proof, se e D.4. Se e also [43 ] an d [66] . 5.15. Corollary . / / K* = K, then the Fredholm operator (5.29 ) has an or- thonormal eigenbasis en (n G N). Furthermore: (a) The eigenvalues a n (n G N) are real and oo (5.33) J2 a n<™- 7 1 = 1 (/?) If the integrals J(t) — f K(t,s) 2 dfi(s) are uniformly bounded, then each function y — ax (y G im a) expands into a (generalized) Fourier series oo (5.34) y(t) = ^2 <*nXne n(t), n=l which converges absolutely and uniformly. Indeed, i f en (n G N) i s an eigenbasi s o f (5.29) , then (5.31 ) implie s tha t oo (5.35) K(t, s) = ^2 a nen(t)en(s). 7 1 = 1 In particular , th e left-han d sid e of (5.33 ) coincide s with ||i^|| 2 , whenc e (a).
- 52. 6. TENSO R PRODUCT S 35 To prov e (/?) , notic e that , fo r a fixed t , th e number s a nen(t) ar e th e Fourie r coefficients o f (5.35) . Applyin g th e Besse l inequality, w e have oo ]T|an en (£)|2 < J(t)<C. n=l Now w e ca n appl y th e Cauchy-Bunyakovsk y inequalit y t o verif y th e convergenc e of series (5.34) . A s a result, we have (/?) , i.e., series (5.34 ) converge s absolutely an d uniformly. Assertion (/? ) i s usually calle d th e Hilbert-Schmidt Theorem. 6. Tenso r product s 6.1. Theorem-Definition . For any pair of vector spaces AT, Y over a field F there exists a unique, up to isomorphism, vector space X ® Y, called the tenso r product o f X an d Y, which is defined by the following conditions. (a) There is a bilinear map n: X x Y —* X 0 Y, (x , y)i — » x ® y, whose image generates X 0 Y. (j3) For any pair of linearly independent systems e , / in (respectively) X , Y the system e® f consisting of vectors x®y, where x G e, y G /, is linearly independent in X ®Y. In addition, as follows from (a) and ((3), X ® Y has the following universal property in VECT^? : (7) Any bilinear map a: X xY — » Z (in VECT^?) has a unique extension to a linear map j3: X ®Y — > Z given by (6.1) a{x,y) = fi(x®y), where x G X and y <EY. Finally, X(&Y is uniquely determined (up to isomorphism) by conditions (a) and (7) . PROOF. Le t F[X xY] b e th e vecto r spac e over F wit h basi s X xY (se e 2.1). We define X<S>Fa s th e quotien t spac e o f F[X x Y] b y th e subspac e N generate d by the element s (6.2) (Xx + fix, y) - (x, y) - /i(x' , y), (6.3) (x , Xy + /V ) - H x i v) - M^ > y'), where A,/iGF , X,X' G X, an d y,y f G Y. B y thi s definition , th e canonica l projec - tion 7r : F[X xY] — > X <S> Y i s bilinear an d cover s X ®Y. Settin g 7r(x , y) = x 0 y, we have (a). Nex t w e want t o sho w that conditio n (7 ) i s satisfied . Indeed, (6.1 ) define s [5 on the basi s 1x 7, whic h gives rise to a linear ma p o n F[X x Y] (se e 2.3) . Sinc e a i s bilinear, w e have tha t (3 annihilates element s (6.2 ) and (6.3) , i.e. , /3(N) = 0 . The n (3 i s defined o n th e coset s (modAQ , i.e. , i t ca n b e viewed a s a linear operato r X 0 Y — » Z. Thi s implie s (7) . In particular , fo r eac h pai r o f linea r operator s a G Hor n (X, X') an d b G Hom(F, Y') (i n VECTi? ) th e ma p a(x,y) = ax ® by i s bilinear , an d therefor e gives rise to a linea r operato r (3 = a 0 6 G Horn (X 0 Y , X' 0 Y') suc h tha t (6.4) (a (8) b)(x ® y) = ax ® by, where x G X an d y G Y. Late r i n th e book , w e shal l discus s th e connectio n between th e symbo l a 0 b and th e tenso r produc t symbo l i n (a). Similarly , fo r an y
- 53. 36 1. BASI C NOTION S pair o f linea r functional s / G X* an d g G Y*, w e ca n defin e a linea r functiona l f®ge{X®YY by (6.5) (f®g){x®y) = f{x)g{x). Here we identify F®F wit h F, s o that (6.5 ) can be viewed as a special case of (6.4). Let e , / b e base s i n (respectively ) X , Y an d le t e,(p b e thei r dua l system s i n (respectively) X* , Y*. Applyin g (6.5) , we have that £®</ ? is a dual system for e®/ . Consequently, e ® / i s linearl y independent , an d therefor e (b y (a)) i s a basi s o f X®Y. Thi s implie s (/?) . As a result, w e have constructed a space X< 8 > Y satisfyin g condition s (a) , (/3) , and (7) . Furthermore, i f Z (i n place of X (8>Y) satisfies condition s (a) an d (/3) , the ma p f3(x ® y) = x <g> y is (b y equalit y o f th e dimensions ) a n isomorphis m X ® Y « Z . Similarly, i f Z satisfie s condition s (a ) an d (7) , then Z « X ® Y. D 6.2. Corollary , (i ) dim( X(8 ) Y) = dim X • dim Y. (ii) For each pair of bases e and f, the elements z G X (&Y can be uniquely written as (6.6) z = y ^2zij{ei® fj), with Zij G F. Indeed, eac h o f the system s e and / ca n b e extende d t o a basi s (respectively , in X an d Y) . Henc e (i ) i s a particular cas e of (/3 ) i n 6.1 . Assertio n (ii ) i s obvious. Note that, fo r th e vector s z = x ® y i n (6.6) , one has z^ — Xiyj. Exercises. 1 . Identifyin g 1 (8) x (respectively , x (g) 1 ) with x G X, w e have (u p to a n isomorphis m i n VECT^ ) (6.7) F®X = X®F = X. 2. Fo r eac h linearly independen t syste m e in X, th e equalit y (6.8) Yl ei ®yi = 0 in X (8 ) Y i s only possibl e when yi = 0 (fo r al l i G / ) . 3. Th e elements x®y exhaus t X® Y onl y when either dimX = 1 or dim Y = 1 . Remark. On e ca n defin e X (g ) Y a s a forma l linea r hul l o f the syste m e (8) /. Theorem 6. 1 then shows that thi s definition i s independent o f the chosen bases e, / in (respectively ) X , Y. 6.3. Definition . Le t Xi (i = 1 ,... , n) b e a finite famil y o f vector space s ove r a field F. Repeatin g th e argument s o f 6.1 , we have a unique (u p t o isomorphism ) tensor produc t n (6.9) X = ® X i = Xi<8>---®X n, 1=1 defined by : (a) a n n-linea r ma p 7r(xi,... , xn) — x (8 ) • • •< 8> xn, whos e imag e generate s X , and th e conditio n n {fi) dim X = I}dimXi.
- 54. 6. TENSO R PRODUCT S 37 In thi s case , conditio n (7 ) mean s tha t an y n-linea r ma p a: X x • • • x X n— > Z ca n be uniquel y extende d t o a linea r ma p (3 X— » Z. In particular , le t n = 3 . Applyin g (7 ) t o th e trilinea r function s a(x,y,z) = x®(y®z) an d a'(x,y, z) = (a;®y)®z , w e hav e tha t th e tenso r produc t ! ® 7 ® Z in VECT V coincide s (u p t o isomorphism ) wit h (6.10) X 0 ( y 0 Z ) = ( X(g ) Y) 0 Z . Relation (6.1 0 ) ca n b e interprete d a s th e associativit y propert y o f tenso r prod - uct. Clearly , thi s multiplicatio n i s distributiv e wit h respec t t o th e direc t su m i n VECTV. Accordin g t o (6.7) , th e one-dimensiona l spac e F i s a n identit y fo r th e tensor produc t (6.9) . Definition (6.9 ) ca n be extended t o arbitrary familie s o f vector space s Xi (i G I). In tha t case , X i s spanne d b y th e element s Xi x 0 • • • 0 Xi n, wher e i, ..., i n i s a n arbitrary (finite ) se t o f indice s i G I. 6.4. Operator s a 0 6 . Th e ma p (a , b) i-» a 0 6 defined i n (6.4 ) i s bilinear, an d therefore extend s t o a linea r ma p (6.11) Hor n (X, X') 0 Hor n (y, Y') - > Hor n (X ^ 7 , 1 ' ® y ' ) , defined b y a 0 f r i - ^ a 0 6 (i n th e left-han d side , th e tenso r symbo l i s define d a s i n 6.1 (a)) . I t i s clea r tha t operator s (6.4 ) satisf y (6.12) (a ' 0 &')( a 0 6 ) = a' a 0 6'b , whenever eac h o f th e composition s a'a, b'b i s defined . Not e th e followin g specia l cases o f (6.1 1 ) : (6.13) En d X 0 E n d F - • En d ( I 0 F ) , (6.14) x * 0 y * - > ( x ® y ) * . 6.5. Proposition . TTi e ma p (6.1 1 ) (an<i ; m particular, (6. 1 3 ) and (6.1 4) ) is injective. If both d i m X and d i m y are finite, then each of the maps (6.1 3 ) and (6.14) is an isomorphism. P R O O F . Le t e ^ (i G I) b e a basi s o f Hor n (y, Y'). Eac h elemen t i n th e left-han d side o f (6.1 1 ) ca n b e writte n a s h — 2_. hi 0 e^ , i with /i 2 G Hom(X, X f ). Suppos e tha t h — 0 a s a n operato r o n X 0 Y. Applyin g (/ 0 l)/i , wher e / G (X')*, t o x 0 ? / we hav e ^2f(htx)ety = 0 for al l x G X, y G y, an d therefor e f{h tx) = 0 (sinc e th e operator s e ^ ar e linearl y independent) fo r al l / G (X')* . Thu s h tx = 0 (se e 2.3) , i.e. , h z = 0 fo r al l z G J . We no w conclud e tha t ma p (6.1 1 ) i s injective . The surjectivit y o f (6.1 3 ) an d (6.1 4 ) i n the finite-dimensional cas e follow s easil y from th e dimensio n count . Mor e precisely , i f m = d i m X an d n = d i m y , the n the dimension s i n bot h side s o f (6.1 3 ) (respectively , (6.1 4) ) i s (mn) 2 (respectively , mn). •
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. provided with holes. This tube is passed through the throat into the stomach, and when the sack has entered the stomach, the poison is pumped through the flexible tube into the brass one. By turning the handle of the pump the gum tube is closed; then it is forced down again, by means of which another valve on the other side of the brass tube opens, and to this another elastic tube is attached, through which the extracted poison is ejected. Then the handle is turned a second time, which closes this valve and opens the first one, leading to the tube that is fixed in the stomach, and the operation is continued until no poison is left in the stomach. Mr. Weiss told me, that some weeks ago, by means of this instrument, his son had saved the life of a girl, who had taken a considerable dose of arsenic in a fit of amorous desperation. The English nobility give, at certain times, in the British institution, Pallmall street, a public exhibition of their collections of precious paintings. Just now there was the king’s collection of paintings from his palace, Carlton-house, because they were about to demolish this palace, and in its place erect an edifice after the model of the Parthenon at Athens, which is calculated to contain the works of English artists. This is a fine idea, and certainly encouraging for the artists of this nation, but it is a pity that it causes the destruction of this elegant palace. The British institution is a building which consists of three large halls, and which receive their light from above. The collection mostly contained paintings of Flemish artists, some English, and a few Italian and French. There were seven pieces by Rubens, amongst which I particularly noticed his own likeness and that of his first wife, finished in the same manner as those in the collection of Mr. Schamp at Ghent, and at Warwick Castle; besides these, a landscape with figures, representing the history of St. George, with the portraits of Queen Henrietta Maria and Charles I. for whom it was done. Seven paintings by Vandyk, among which the portraits of Charles I. in three views, which his lady had sent to the statuary Bernini at Rome, to finish the bust of the king therefrom. A sketch, studies of horses and horsemen, of remarkable value, and a full-size portrait of Gaston de France, and two portraits of Queen
- 57. Henrietta Maria, which, like that of her unfortunate husband, I might call unavoidable, because it is to be found almost in every collection of paintings in England. I found seven pieces by Rembrandt, among which were several excellent portraits, and his own; they were all easily distinguished by his particular colouring. Fourteen paintings by Teniers, collections of people; small portraits; a view of the towns of Holland, and a couple of landscapes, one of which represents likewise, the artist, his wife, and his gardener; a real ornament to this collection. One of these pieces, representing a village festival, had been on the artist’s harpsichord. I admired two other pieces, in the same style, by J. Ostade, and seven by A. Ostade; six by Jan Steen. One of the latter, very excellently finished, represented an elderly man, just rising from bed, who is listening to the reproaches of a young girl, for his niggardliness; she holds forth to him a trifle of money, and an old woman is urging him to be more generous. Four effects of light, by Schalken, and a portrait by Holbein, are likewise worthy of attention. Nine pieces by Wouverman are easily distinguished by the white horses, representing skirmishes and country scenes. Seven pictures by Mieris are to be known by their fine keeping. Three pieces are by G. Douw, one by Slingelandt, and five by Metzu. A landscape by Ruisdael, and two by Hobbema, attracted my particular attention, as well as eleven pieces by Vandevelde, representing sea-pieces, landscapes, and views of several cities of Holland; two of the latter are finished by him and Vanderheyden jointly; I observed likewise, four very fine pieces by Vanderwerff, one of them representing the Roman Mercy, the other a concert, the third Lot with his daughters, and the fourth two children. The collection is likewise rich in paintings of animals; there are four capital works by G. Potter, one of them representing two hogs, as true as if they were living. A piece by Hondekoeter, representing a chicken, belongs likewise to this class, as well as ten pieces by Cuyp, in which the landscapes are very well finished. Among these I enjoyed particularly a camp-scene with a horseman in the fore- ground, engaged in currying his horse. Six very good pieces, by
- 58. Berghem ought not to be omitted, nor a handsome sea-piece by Buckhuizen, with a view of Briel in the back-ground. Besides these pieces, there is a good collection of other paintings of the same school, but it would lead me too far, to mention them all. From other schools there are but a few and of less value. A landscape with sheep, by Titian; Christ taken down from the Cross, by Michael Angelo and Venusti, and another piece by Gonzalez. Among the paintings of modern times, I found the portrait of Garrick and his wife, by Hogarth; a domestic scene, by Greuze, and several pieces of an Italian painter, Zeffani. One of them, which has become more generally known by the copperplate of Bartolozzi, represents the Royal Academy of London, and the other the Gallery of Florence, with the portraits of several Englishmen of note, who sojourned at that time in Florence. By the same artist I saw two pieces, representing the interior of two royal palaces, with the children of George III. and their mother. These tasteless pieces, compared with the before-mentioned elegant paintings, make an unpleasant impression. I saw eight pieces by Sir Joshua Reynolds, among which his own portrait and two full-size portraits of the Portuguese Chief Marshal Count von der Lippe, and of the English General Marquis of Granby. These two pieces are masterly works, and full of expression. You distinguish in the countenance of Count von der Lippe and in his whole posture, his profound and enterprising spirit, and in the features of Lord Granby his great benevolence, which procured him in the army the name of the soldier’s friend. The features of the count excite respect, while those of the lord claim your attachment. I was much less pleased with the historical pieces of Sir Joshua. The most handsome of the newer paintings was undoubtedly the interior of the choir of a Capuchin chapel by the French painter Granet. The expression of the countenances of the monks is unparalleled; in some you see piety, in others listlessness; another couple make sport of the exceeding piety of a monk, kneeling in the middle of the hall; the countenance of a young, tall, stout monk, is the personification of fanaticism. Near the altar stands a monk in the sacerdotal habit, with two choristers and
- 59. tapers in their hands, the monk singing a hymn. The light is very well executed; it enters through a large window in the back ground, and makes a fine effect on the bare crown of the head and the gray beard of the priest. I think this piece one of the finest of the whole collection. I saw here a great many gentlemen and ladies, and it is said to be fashionable to visit this splendid gallery in the afternoon. On the 26th of July, (the anniversary of the day on which I first landed on American ground at Boston,) I went to the custom-house for the purpose of taking passage for Ostend on board the steam- boat Earl of Liverpool, Captain Peak, which was laying there at anchor. At the custom-house I was quite surprised. I expected to see the splendid, newly-erected palace for the offices of the custom- house, the same which, three years ago, I had admired so much, and instead of it, found nothing but ruins. They said that the foundation had not been well enough examined upon which the custom-house had been built by contract; the building cracked, the large, splendid hall was near falling down, and in order to prevent this accident, they were obliged to demolish the centre building; both wings of the building were yet supported by beams, but they soon will have to demolish them likewise, in order to build an entirely new house. The gentleman who made the contract to have the house built, lays the blame of this bad work upon the architect, and he upon a commission, under whose control he acted. The Earl of Liverpool, of one hundred and thirty tons, with two engines, left London at eleven o’clock, A. M., and on the next morning at six o’clock I landed at Ostend. At four o’clock, P. M. I proceeded by the way of Bruegge to Ghent. During this journey I remembered an observation which I had heard frequently in America, that upon an American visiting Europe for the first time, nothing makes a greater impression than the old monuments, which trace the time past for many centuries, and which are a proof of the prosperity and good taste of preceding generations. I found this observation perfectly true, by my own feelings on returning from America, which exhibits none but new objects, and has nothing but a bustling present struggling for future improvement.
- 60. On the 28th of July, at four o’clock, P. M., I arrived at Ghent.
- 61. —Trans. II.1 Accommodation is here so difficult to procure, that the senators are obliged to sleep three upon one mattress laid upon the floor: their food consists, it is said, almost without exception of salted pork. II.2 In this part of the country, they have either feather beds or moss mattresses; if these latter are old, the moss clots together, and it is like lying on cannon-balls. II.3 [This is the same corps which the Philadelphians extol so highly, that one might almost suppose them equal to the artists of the Theatre Français, if, unfortunately, one visit to the theatre, did not completely dispel the illusion!] II.4 Among the slave traders, a Hollander from Amsterdam, disgusted me particularly, his name was Jacobs. He had the most vulgar and sinister countenance imaginable, was constantly drunk, and treated the wretched negroes in the most brutal manner; he was, however, severely beaten by these miserable beings, driven to despair. II.4a II.4a The virtuous indignation of the Duke, at these horrible consequences of slavery, is such as every man, not hardened by long familiarity with such scenes, must feel; those to whom they are daily presented regard them with calm indifference, or even attempt to argue in favour of their continuance and harmlessness. It is not as generally known, as it should be, that the slave trade is carried on, almost as vigorously now, as ever it was, and by citizens of almost every nation; not in the least excepting Americans. The slave vessels sail principally from Havanna and St. Thomas, and land their cargoes on the island of Puerto Rico, and elsewhere, whither purchasers and agents resort, when such
- 62. —Trans. an arrival occurs. Two schooners, with large cargoes, arrived in Puerto Rico in February last, and two brigs were daily expected. It is said in the West Indies, that all ships of war, of powers owning West India Colonies, connive at the trade, which is fully supported by facts; as French, Danish, and English cruisers were in the vicinity, when the above mentioned cargoes arrived. The idea of cruising off the coast of Africa, to prevent the trade, is ridiculed by the slave dealers, with one of whom the writer of this note conversed. If the American, or any other government really wished to put an end to this trade, it could be very effectually accomplished, by sending small armed vessels to intercept the slave traders near their places of landing cargoes, which are not very numerous. It is also said, in the West Indies, that the Havanna traders still contrive to introduce Africans into the southern part of the United States; of the truth or falsehood of this, we know nothing. The slave vessels are generally Baltimore clipper brigs, and schooners, completely armed and very fast sailers. Two of them sailed on this execrable trade in February last, from a part visited by the writer.—Trans. II.5 Colonel Croghan was one of the most distinguished officers of the American army. In the last war, he defended a miserable stockade, (Fort Stephenson) on Lake Erie, against a force eight times greater than his own, which had artillery, and drove it back. After the peace, he was appointed Post Master of New Orleans, and during my stay in this city, he had entered again into the army, and held the post of second Inspector- General. II.6 [Commonly known in New Orleans by the name of the Calaboose, (from Calabozo, the Spanish term for a vaulted dungeon,) and a great terror to evil-doers in that city; the efficiency of the police of which can never be enough admired.] II.7 I have already made some remarks with regard to the apprehension of negroes in Charleston. If a person wishes here to have a house-negro, male or female,
- 63. —Trans. chastised, they are sent with a note, in which the number of lashes which the bearer is to receive are specified, with a quarter of a dollar; he or she is lodged in the slave prison. Here the slave receives the punishment, and a certificate, which he must carry to his master. The maximum of lashes is thirty-nine, according to the Mosaic law. The species of punishment is specified as in Charleston, or “aux quatre piquets.” In this last case, the poor wretch is pressed out flat on his face upon the earth, and his hands and feet bound to four posts. In this posture he receives his flogging. This frightful method of chastisement, is principally in use on the plantations; and cruel discipline is there chiefly practised. Whoever wishes to punish a house servant severely, either hires or sells him to work on the plantations. II.8 If it be known that a stranger, who has pretensions to mix with good society, frequents such balls as these, he may rely upon a cold reception from the white ladies. II.9 [A plain, unvarnished history of the internal slave trade carried on in this country, would shock and disgust the reader to a degree that would almost render him ashamed to acknowledge himself a member of the same community. In unmanly and degrading barbarity, wanton cruelty, and horrible indifference to every human emotion, facts could be produced worthy of association with whatever is recorded of the slave trade in any other form. One of these internal slave traders has built, in a neighbouring city, a range of private prisons, fronting the main road to Washington, in which he collects his cattle previous to sending off a caravan to the south. The voice of lamentation is seldom stilled within these accursed walls.]
- 64. —Trans. —Trans. —Trans. II.10 This Frenchman, a merchant’s clerk from Montpelier, was not satisfied with this: he went to the police, lodged a complaint against the girl, had her arrested by two constables, and whipped again by them in his presence. I regret that I did not take a note of this miscreant’s name, in order that I might give his disgraceful conduct its merited publicity. II.11 [Nonsense.] II.12 [Our author has somehow been confused in his diary here: the mouth of La Fourche is generally called seventy-five miles above New Orleans, Stoddart makes it eighty-one. At any rate it is about half way between Bayou Sara or Point Coupee and the city of New Orleans; and of course the Duke must have passed Donaldsonville, which is at the junction of La Fourche with the Mississippi, in the morning of the day he passed Baton Rouge.] II.13 In these rivers there is a difference understood between the two kinds of trunks of trees which lie in the stream, and are dangerous to vessels, i. e. snags and sawyers. The first, of which I have spoken already in the Alabama river, are fast at one end in the bottom, and stand up like piles; the others are not fastened, by being moved by the current the upper end of the tree takes a sawing motion, from whence its appellation is derived. II.14 Coluber coccineus. II.15 [These log turnpikes are better known by the name of “corduroy roads.”] II.16 [This is, perhaps, the most charitable idea that can be formed of the actions of such reformers, as well as of a “lady” heretofore mentioned, who has unsexed herself, and become so intoxicated with vanity, as
- 65. —Trans. —Trans. —Trans. —Trans. —Trans. —Trans. —Trans. —Trans. enthusiastically to preach up a “reformation” in favour of the promiscuous intercourse of sexes and colours, the downfall of all religion, and the removal of all restraints imposed by virtue and morality!] II.17 [It is understood that Mr. M‘Clure has long since given up all connexion with the New Harmony bubble.] II.18 By late newspapers it appears, that the society actually dissolved itself, in the beginning of the year 1827. II.19 [According to the report of some females, who were induced to visit New Harmony, and remained there for some time, any situation much above abject wretchedness, was preferable to this vaunted terrestrial paradise.] II.20 He was drowned in the Wabash, which he attempted to swim over on horseback. II.21 [He is at this time advertising a boarding-school in the Western country, on his own account, which is to be under his immediate superintendence!] II.22 [He has left it some time since, as well as Dr. Troost.] II.23 These had been presented to Bishop Fenwick by Cardinal Fesch, for his cathedral, and were only here, until they could find their place in the Temple of God. II.24 [Brother of Lord Hill.] II.25 [Peyton Symmes, Esq. receiver of the land office.] II.26 [Tetrao Umbellus, L.] II.27 These meadows are designated in America, by the name of prairies, and extend over large tracts of
- 66. —Trans. —Trans. land in the western country; they are covered with high grass; trees grow very sparingly on them, while the surrounding forests exhibit the most beautiful trees; the soil of these prairies generally consists of turf-moor. II.28 [The Ohio Eagle.] II.29 Knopendraayerye. II.30 In the year 1826, I enjoyed three springs; the first about the end of February at New Orleans, the second at New Harmony and Louisville, and the third in the state of Ohio, and west Pennsylvania. II.31 In this hymn-book are some pieces, which, if the perfect child-like innocence of these maidens be not recollected, might appear rather scandalous. For instance, there is a literal translation of the song of Solomon, among others. II.32 [It is to be hoped that the able and luminous report of the commissioners appointed by the state, to make inquiries on the subject of penitentiary discipline, will be sufficient to correct the glaring errors of this new system; which like most of the new systems of the present day, is clearly proved thereby to be more specious than beneficial. The evidence accumulated by the commissioners is of a character to satisfy every candid mind, not chained to the support of a particular theory, that solitary confinement without labour, is unequal in operation, inadequate to the end proposed, and promises to be as destructive to human life as it is discordant to humane feelings.] II.33 [This is a very erroneous idea. The taste for painting and music has not been cultivated, generally, in this country, on account of the condition of property and society, which demand of Americans a primary
- 67. —Trans. —Trans. —Trans. —Trans. devotion to things absolutely necessary and useful. As wealth becomes more accumulated, artists will be encouraged; and then we have no fear of their being long inferior to the artists of any other nation.] II.34 [The reason is, that the portrait painter ministers to the gratification of personal vanity, or self-love, and the landscape painter to a refined taste. As the proportion of egotists to men of refined and cultivated taste, is somewhat less than a million to one, it is easy to see which branch of the arts will receive most attention.] II.35 Kensington was formerly a distinct village, on the Delaware above Philadelphia; the city has now extended thus far, so that it now belongs to the city. The tree was some years ago struck by lightning and destroyed. II.36 In service of the Netherlands, on his return from a mission to the new South American republics. II.37 Austrian Consul-General to the United States. II.38 Whose acquaintance I made on the Mississippi during my trip from Louisville to Cincinnati. II.39 [This light is emitted by molluscous animals, which are exceedingly abundant in some parts of the ocean. They are also seen to great advantage during the night, in the Chesapeake bay.] II.40 A respectable London merchant, and native of Flanders, to whom I am much indebted for very important services. II.41 [Now exhibiting in New York.]
- 68. THE END.
- 69. Spelling and Typography Spelling was corrected if the mistake was clearly mechanical, or inconsistent with the author’s (or translator’s) usage elsewhere. This includes some spellings that were acceptable in 1828, but are different from other occurrences of the same word. All commas are as printed. Inconsistent italicization of ships’ names is as in the original. The notation “invisible” means that there is an appropriately sized empty space, but the punctuation itself is missing. Some specifics: “Bodleïan” is written with dieresis “chesnut” is standard for the time and is used consistently “team boat” (referring to canals) is not an error “lime-stone” and “sand-stone” are hyphenated at the beginning of the book but later become single words “free-stone” or “freestone” are not frequent enough to establish a pattern “country seat” starts out as two words, but later becomes hyphenated “country-seat” the inconsistent spacing of “no( )where” “every( )where” and “else( )where” is unchanged “back ground” is generally two words; “fore-ground” (with hyphen) occurs only once Spelling (unchanged): appointed him his aid present as aid to the Emperor Mr. Butler, his aid [the spelling aid is used consistently] the tatoed and dried head of a New Zealand chief roast-beef, plumb-pudding, &c. This place is called the antichamber. it is fixed in a cramp [text unchanged: error for clamp?] some negroes, who were frolicing during the Christmas holy-days corset inventress to the Dutchess of Kent [Duchess and Dutchess are each used once] French (corrected):
- 70. From the left wing a line runs en crémaillère [crémaillére] When a lady is left sitting, she is said to be “bredouille.” [bredouillè] so that they would consider their labour in the light of a corvée. [corveè] Errors (corrected): strange bas-reliefs, representing ancient hunting scenes [bass- reliefs] the rooms are not large; the beds [is not] On the top of the capitol is a cupola [cupalo] the pen is too feeble to delineate the simultaneous feelings of insignificance and grandeur [simultaneons ... insignificence] then a company of sappers and miners, [minors] in which again each sex has its own side [each sect] There are two paintings by Teniers [painting] the office and place of deposit for bound bibles [deposite] They recall to memory Glenn’s Falls on the Hudson [recal; recall is used consistently elsewhere] the other loses by faint colouring [looses] an English copy of the illustrations of Göthe’s Faust laid open. [illustratrations; spelling Göthe with umlaut unchanged] In a hollow place there is a basin, or rather a reservoir [their is] close by it stands the prison, or county gaol, [goal] this journey of one hundred and ninety-eight miles [ninty-eight] the log houses were only employed as negro cabins [onegr] we might lie several days, perhaps weeks here [several day] because they had not received their pay for some time. [missing not] Several of the French families here settled [familes] so as not to lose themselves in the woods [loose] and on this account, the proceeding to me appeared arbitrary [acount] has followed me even in America like an evil genius [and evil] an inspection on the Red river, the Arkansas, and New Orleans [Arkansa; Arkansas is used consistently elsewhere] a great part of the houses are built of brick; [missing a] We frequently rode along the new national turnpike road [frequently road] over some stone bridges of sumptuous construction [contruction]
- 71. about the right bank of the Alleghany and Ohio [righ] parallel ridges, called Laurel hill. [callel] the girls learn to sew and knit [sow] The gentlemen above named accompanied me to the vessel. [accompaned] every thing, manufactured in Birmingham, [Burmingham] a fine view into a considerable suite of rooms [considerble] until no poison is left in the stomach [stomuch] Spacing, hyphenization, capitalization: By his highness, Bernhard, Duke of Saxe-Weimar Eisenach. [capitalized as shown] two stories besides a ground floor, and may contain [floor,and] so that it is excellently adapted to waterworks [anomalous missing hyphen unchanged] They have a large kitchen garden [they have] The corpse is put in the corpse-house [the corpse] views of Monticello, Mount Vernon, the principal buildings in Washington [Mount-Vernon] many evergreen trees and bushes. [ever green] with Madam Herries; he is a Frenchman [Herries;he] This is, however, the case with most of the stores [this is] Punctuation: Footnote I-4: [This manuscript .... of London.]—Trans. [printed of London.—Trans.]: changed for consistency] On the 9th, at 6 A. M. she arrived at Falmouth. [6 A. M] Schenectady.— Utica.— Rochester. [—Utica—] and produce much vexation in consequence of the baggage. [final . missing] the village of St. Regis, the last belonging to the United States. [final . missing] a monument erected by the colony in honour of Lord Nelson. [final . missing] to protect the place of embarkation by a fort. [final . missing] On the ensuing morning I went with Mr. Halbach to Mr. Vaux [Mr Vaux: period invisible] mineralogy and geology.— ... lectures on chemistry. [missing . after geology and chemistry] drawing of the human figure.— [final . missing]
- 72. The cotton cleaned from its seed is put into a large chest, pressed in, and packed up. [final . missing] Mr. Nott studied in England and France [Mr Nott: period invisible] the 16th ultimo from Liverpool [ultimo.] [Footnote II-9: [... within these accursed walls.]—Trans. [missing — before Trans.] the river is fordable in many places above the falls.” [close quote missing] a diameter of one hundred and fifty yards during forty miles.” [close quote missing] even this navigation so expensive and destructive to the wood, will cease [punctuation unchanged] one hundred and twenty horses, which daily work here, [work here.] in the neighbourhood of Lake Ontario and the river St. Lawrence [St Lawrence]
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