![Chapter 1
Some notions from functional
analysis
1.1 Vector and normed spaces
The notion of a vector space is obtained by axiomatization of the properties of
the three–dimensional space of Euclidean geometry, or of configuration spaces of
classical mechanics. A vector (or linear) space V is a set {x, y, . . . } equipped
with the operations of summation, [x, y] → x + y ∈ V , and multiplication by a
complex or real number α, [α, x] → αx ∈ V , such that
(i) The summation is commutative, x + y = y + x, and associative, (x + y) + z =
x+(y +z). There exist a zero element 0 ∈ V , and an inverse element −x ∈ V,
to any x ∈ V so that x + 0 = x and x + (−x) = 0 holds for all x ∈ V .
(ii) α(βx) = (αβ)x and 1x = x.
(iii) The summation and multiplication are distributive, α(x + y) = αx + αy and
(α + β)x = αx + βx.
The elements of V are called vectors. The set of numbers (or scalars) in the definition
can be replaced by any algebraic field F. Then we speak about a vector space over F,
and in particular, about a complex and real vector space for F = C, R, respectively.
A vector space without further specification in this book always means a complex
vector space.
1.1.1 Examples: (a) The space Cn
consists of n–tuples of complex numbers with
the summation and scalar multiplication defined componentwise. In the same
way, we define the real space Rn
.
(b) The space p
, 1 ≤ p ≤ ∞, of all complex sequences X := {ξj}∞
j=1 such that
∞
j=1 |ξj|p
∞ for p ∞ and supj |ξj| ∞ if p = ∞, with the summation
and scalar multiplication defined as above; the Minkowski inequality implies
X + Y ∈ p
for X, Y ∈ p
(Problem 2).
1](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-24-320.jpg)
![2 1 Some notions from functional analysis
(c) The space C(J) of continuous complex functions on a closed interval J ⊂ R
with (αf +g)(x) := αf(x)+g(x). In a similar way, we define the space C(X)
of continuous functions on a compact X and spaces of bounded continuous
functions on more general topological spaces (see the next two sections).
A subspace L ⊂ V is a subset, which is itself a vector space with the same
operations. A minimal subspace containing a given subset M ⊂ V is called the
linear hull (envelope) of M and denoted as Mlin or lin(M). Vectors x1, . . . , xn ∈ V
are linearly independent if α1x1 + · · · + αnxn = 0 implies that all the numbers
α1, . . . , αn are zero; otherwise they are linearly dependent, which means some of them
can be expressed as a linear combination of the others. A set M ⊂ V is linearly
independent if each of its finite subsets consists of linearly independent vectors.
This allows us to introduce the dimension of a vector space V as a maxi-
mum number of linearly independent vectors in V . Among the spaces mentioned in
Example 1.1.1, Cn
and Rn
are n–dimensional (Cn
is 2n–dimensional as a real vec-
tor space) while the others are infinite–dimensional. A basis of a finite–dimensional
V is any linearly independent set B ⊂ V such that Blin = V ; it is clear that
dim V = n iff V has a basis of n elements. Vector spaces V, V
are said to
be (algebraically) isomorphic if there is a bijection f : V → V
, which is linear,
f(αx + y) = αf(x) + f(y). Isomorphic spaces have the same dimension; for finite–
dimensional spaces the converse is also true (Problem 3).
There are various ways to construct new vector spaces from given ones. Let us
mention two of them:
(i) If V1, . . . , VN are vector spaces over the same field; then we can equip the
Cartesian product V := V1 × · · · × VN with a summation and scalar multipli-
cation defined by α[x1, . . . , xN ]+[y1, . . . , yN ] := [αx1 +y1, . . . , αxN +yN ]. The
axioms are obviously satisfied; the resulting vector space is called the direct
sum of V1, . . . , VN and denoted as V1 ⊕ · · · ⊕ VN or
⊕
j Vj. The same term
and symbols are used if V1, . . . , VN are subspaces of a given space V such
that each x ∈ V has a unique decomposition x = x1 + · · · + xN , xj ∈ Vj.
(ii) If W is a subspace of a vector space V , we can introduce an equivalence
relation on V by x ∼ y if x−y ∈ W. Defining the vector–space operations on
the set Ṽ of equivalence classes by αx̃+ỹ := (αx+y)˜ for some x ∈ x̃, y ∈ ỹ,
we get a vector space, which is called the factor space of V with respect to
W and denoted as V/W.
1.1.2 Example: The space Lp
(M, dµ) , p ≥ 1, where µ is a non–negative measure,
consists of all measurable functions f : M → C satisfying
M
|f|p
dµ ∞ with
pointwise summation and scalar multiplication — cf. Appendix A.3. The subset
L0 ⊂ Lp
of the functions such that f(x) = 0 for µ–almost all x ∈ M is easily
seen to be a subspace; the corresponding factor space Lp
(M, dµ) := Lp
(M, dµ)/L0
is then formed by the classes of µ–equivalent functions.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-25-320.jpg)
![1.2 Metric and topological spaces 7
If µ is σ–finite and µ(M) = ∞, there is a disjoint decomposition ∞
j=1 Mj = M
with µ(Mj) ∞. The already proven completeness of Lp
(Mj, dµ) implies the
existence of functions f(j)
∈ Lp
(Mj, dµ) which fulfil f
(j)
n −f(j)
p → 0 as n → ∞;
then we can proceed as in the proof of completeness of p
(cf. Problem 23).
Other examples of complete metric spaces are given in Problem 23. Any metric
space can be extended to become complete: a complete space (X
,
) is called the
completion of (X, ) if (i) X ⊂ X
and
(x, y) = (x, y) for all x, y ∈ X, and (ii)
the set X is everywhere dense in X
(this requirement ensures minimality — cf.
Problem 25).
1.2.2 Theorem: Any metric space (X, ) has a completion. If (X̃, ˜
) is another
completion of (X, ), there is an isometry f : X
→ X̃ which preserves X, i.e.,
f(x) = x for all x ∈ X.
Sketch of the proof: Uniqueness follows directly from the definition. Existence is
proved constructively by the so–called standard completion procedure which genera-
lizes the Cantor construction of the reals. We start from the set of all Cauchy
sequences in (X, ). This can be factorized if we set {xj} ∼ {yj} for the sequences
with limj→∞ (xj, yj) = 0. The set of equivalence classes we denote as X∗
and define
∗
([x], [y]) := limj→∞ (xj, yj) to any [x], [y] ∈ X∗
. Finally, one has to check that
this definition makes sense, i.e., that ∗
does not depend on the choice of sequences
representing the classes [x], [y], ∗
is a metric on X∗
, and (X∗
, ∗
) satisfies the
requirements of the definition.
The notion of topology is obtained by axiomatization of some properties of
metric spaces. Let X be a set and τ a family of its subsets which fulfils the following
conditions (topology axioms):
(t1) X ∈ τ and ∅ ∈ τ.
(t2) If I is any index set and Gα ∈ τ for all α ∈ I; then α∈I Gα ∈ τ.
(t3) n
j=1 Gj ∈ τ for any finite subsystem {G1, . . . , Gn} ⊂ τ.
The family τ is called a topology, its elements open sets and the set X equipped
with a topology is a topological space; when it is suitable we write (X, τ).
A family of open sets in a metric space (X, ) is a topology by definition;
we speak about the metric–induced topology τ, in particular, the norm–induced
topology if X is a vector space and is induced by a norm. On the other hand,
finding the conditions under which a given topology is induced by a metric is a
nontrivial problem (see the notes). Two extreme topologies can be defined on any
set X: the discrete topology τd := 2X
, i.e., the family of all subsets in X, and the
trivial topology τ0 := {∅, X}. The first of them is induced by the discrete metric,
d(x, y) := 1 for x=y, while (X, τ0) is not metrizable unless X is a one–point set.
An open set in a topological space X containing a point x or a set M ⊂ X is
called a neighborhood of the point X or the set M, respectively. Using this concept,](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-30-320.jpg)
![8 1 Some notions from functional analysis
we can adapt to topological spaces most of the “metric” definitions presented above,
as well as some simple results such as those of Problems 17a, c, 19b, topological
equivalence of homeomorphic spaces, etc. On the other hand, equally elementary
metric–space properties may not be valid in a general topological space.
1.2.3 Example: Consider the topologies τfin and τcount on X = [0, 1] in which
the closed sets are all finite and almost countable subsets of X, respectively. If
{xn} ⊂ X is a simple sequence, xn = xm for n = m ; then any neighborhood U(x)
contains all elements of the sequence with the exception of a finite number; hence
the limit is not unique in (X, τfin). This is not the case in (X, τcount) but there only
very few sequences converge, namely those with xn = xn0 for all n ≥ n0, which
means, in particular, that we cannot use sequences to characterize local continuity
or points of the closure.
Some of these difficulties can be solved by introducing a more general notion of
convergence. A partially ordered set I is called directed if for any α, β ∈ I there
is γ ∈ I such that α ≺ γ and β ≺ γ. A map of a directed index set I into a
topological space X, α → xα, is called a net in X. A net {xα} is said to converge
to a point x ∈ X if to any neighborhood U(x) there is an α0 ∈ I such that
xα ∈ U(x) for all α α0. To illustrate that nets in a sense play the role that
sequences played in metric spaces, let us mention two simple results the proofs of
which we leave to the reader (Problem 29).
1.2.4 Proposition: Let (X, τ) , (X
, τ
) be topological spaces; then
(a) A point x ∈ X belongs to the closure of a set M ⊂ X iff there is a net
{xα} ⊂ M such that xα → x.
(b) A map f : X → X
is continuous at a point x ∈ X iff the net {f(xα)}
converges to f(x) for any net {xα} converging to x.
Two topologies can be compared if there is an inclusion between them, τ1 ⊂ τ2,
in which case we say that τ1 is weaker (coarser) than τ2; while the latter is stronger
(finer) than τ1. Such a relation between topologies has some simple consequences
— see, e.g., Problem 32. In particular, continuity of a map f : X → Y is preserved
when we make the topology in Y weaker or in X stronger. In other cases it may
not be preserved; for instance, Problem 3.9 gives an example of three topologies,
τw ⊂ τs ⊂ τu, on a set X := B(H) and a map f : X → X which is continuous with
respect to τw and τu but not τs.
1.2.5 Example: A frequently used way to construct a topology on a given X
employs a family F of maps from X to a topological space (X̃, τ̃). Among all
topologies such that each f ∈ F is continuous there is one which is the weakest; its
existence follows from Problem 30, where the system S consists of the sets f(−1)
G)
for each G ⊂ τ̃, f ∈ F. We call this the F–weak topology.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-31-320.jpg)
![1.2 Metric and topological spaces 9
For any set M in a topological space (X, τ) we define the relative topology τM
as the family of intersections M ∩ G with G ⊂ τ; the space (M, τM ) is called
a subspace of (X, τ). Other important notions are obtained by axiomatization of
properties of open balls in metric spaces. A family B ⊂ τ is called a basis of a
topological space (X, τ) if any nonempty open set can be expressed as a union of
elements of B. A family Bx of neighborhoods of a given point x ∈ X is called a
local basis at x if any neighborhood U(x) contains some B ∈ Bx. A trivial example
of both a basis and a local basis is the topology itself; however, we are naturally
more interested in cases where bases are rather a “small part” of it. It is easy to see
that local bases can be used to compare topologies.
1.2.6 Proposition: Let a set X be equipped with topologies τ, τ
with local bases
Bx, B
x at each x ∈ X. The inclusion τ ⊂ τ
holds iff for any B ∈ Bx there is
B
∈ B
x such that B
⊂ B.
To be a basis of a topology or a local basis, a family of sets must meet certain
consistency requirements (cf. Problem 30c, d); this is often useful when we define a
particular topology by specifying its basis.
1.2.7 Example: Let (Xj, τj), j = 1, 2, be topological spaces. On the Cartesian
product X1 × X2 we define the standard topology τX1×X2 determined by τj, j =
1, 2, as the weakest topology which contains all sets G1 × G2 with Gj ∈ τj, i.e.,
τX1×X2 := τ(τ1 × τ2) in the notation of Problem 30b. Since (A1 × A2) ∩ (B1 × B2) =
(A1 ∩ B1) × (A2 ∩ B2), the family τ1 × τ2 itself is a basis of τX1×X2 ; a local basis
at [x1, x2] consists of the sets U(x1) × V (x2), where U(x1) ∈ τ1, V (x2) ∈ τ2 are
neighborhoods of the points x1, x2, respectively. The space (X1 × X2, τX1×X2 ) is
called the topological product of the spaces (Xj, τj), j = 1, 2.
Bases can also be used to classify topological spaces by the so–called countability
axioms. A space (X, τ) is called first countable if it has a countable local basis at
any point; it is second countable if the whole topology τ has a countable basis. The
second requirement is actually stronger; for instance, a nonseparable metric space
is first but not second countable (cf. Problem 18; some related results are collected
in Problem 31). The most important consequence of the existence of a countable
local basis, {Un(x) : n = 1, 2, . . .} ⊂ τ, is that one can pass to another local basis
{Vn(x) : n = 1, 2, . . .}, which is ordered by inclusion, Vn+1 ⊂ Vn, setting V1 := U1
and Vn+1 := Vn ∩ Un+1. This helps to partially rehabilitate sequences as a tool in
checking topological properties (Problem 33a).
The other problem mentioned in Example 1.2.3, namely the possible nonunique-
ness of a sequence limit, is not related to the cardinality of the basis but rather to the
degree to which a given topology separates points. It provides another classification
of topological spaces through separability axioms:
T1 To any x, y ∈ X, x=y, there is a neighborhood U(x) such that y ∈ U(x).
T2 To any x, y ∈ X, x=y, there are disjoint neighborhoods U(x) and U(y).](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-32-320.jpg)
![1.4 Topological vector spaces 13
α := supx∈X f(x) and let {xn} ⊂ X be a sequence such that f(xn) → α. Since X
is compact there is a subsequence {xkn } converging to some xs and the continuity
implies f(xs) = α. In the same way we can check that f assumes a minimum value.
1.4 Topological vector spaces
We can easily check that the operations of summation and scalar multiplication
in a normed space are continuous. Let us now see what would follow from such a
requirement when we combine algebraic and topological properties. A vector space
V equipped with a topology τ is called a topological vector space if
(tv1) The summation maps continuously (V × V, τV ×V ) to (V, τ).
(tv2) The scalar multiplication maps continuously (C×V, τC×V ) to (V, τ).
(tv3) (V, τ) is Hausdorff.
In the same way, we define a topological vector space over any field. Instead of
(tv3), we may demand T1–separability only because the first two requirements imply
that T3 is valid (Problem 39).
A useful tool in topological vector spaces is the family of translations,
tx : V → V , defined for any x ∈ V by tx(y) := x + y. Since t−1
x = t−x, the
continuity of summation implies that any translation is a homeomorphism; hence
if G is an open set, then x + G := tx(G) is open for all x ∈ V ; in particular, U
is a neighborhood of a point x iff U = x + U(0), where U(0) is a neighborhood
of zero. This allows us to define a topology through its local basis at a single point
(Problem 40).
Suppose a map between topological vector spaces (V, τ) and (V
, τ
) is simul-
taneously an algebraic isomorphism of V, V
and a homeomorphism of the corre-
sponding topological spaces, then we call it a linear homeomorphism (or topological
isomorphism). As in the case of normed spaces (cf. Problem 21), the structure of a
finite–dimensional topological vector space is fully specified by its dimension.
1.4.1 Theorem: Twofinite–dimensionaltopologicalvectorspaces, (V, τ) and (V
, τ
),
are linearly homeomorphic iff dim V = dim V
. Any finite–dimensional topological
vector space is locally compact.
Proof: It is sufficient to construct a linear homeomorphism of a given n–dimensional
(V, τ) to Cn
. We take a basis {e1, . . . , en} ⊂ V and construct f : V → Cn
by
f
n
j=1 ξjej
:= [ξ1, . . . , ξn]; in view of the continuity of translations we have to
show that f and f−1
are continuous at zero. According to (tv1), for any U(0) ∈ τ
we can find neighborhoods Uj(0) such that
n
j=1 xj ∈ U(0) for xj ∈ Uj(0) , j =
1, . . . , n and f−1
is continuous by Problem 42a. To prove that f is continuous
we use the fact that V is Hausdorff: Proposition 1.3.1 and Theorem 1.3.3 together
with the already proven continuity of f−1
ensure that Sε := {x∈V : f(x) = ε}
= f(−1)
(Kε) is closed for any ε 0; we have denoted here by Kε the ε–sphere](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-36-320.jpg)
![14 1 Some notions from functional analysis
in Cn
. Since 0 ∈ Sε, the set G := V Sε is a neighborhood of zero, and by
Problem 42b there is a balanced neighborhood U ⊂ G of zero; this is possible only
if f(x) ε for all x ∈ U.
Next we want to discuss a class of topological vector spaces whose properties are
closer to those of normed spaces. In distinction to the latter the topology in them is
not specified generally by a single (semi)norm but rather by a family of them. Let
P := {pα : α ∈ I} be a family of seminorms on a vector space V where I is an
arbitrary index set. We say that P separates points if to any nonzero x ∈ V there
is a pα ∈ P such that pα(x) = 0. It is clear that if P consists of a single seminorm
p it separates points iff p is a norm. Given a family P we set
Bε(p1, . . . , pn) := { x ∈ V : pj(x) ε , j = 1, . . . , n } ;
the collection of these sets for any ε 0 and all finite subsystems of P will be
denoted as BP
0 . In view of Problem 40, BP
0 defines a topology on V which we
denote as τP
.
1.4.2 Theorem: If a family P of seminorms on a vector space V separates points,
then (V, τP
) is a topological vector space.
Proof: By assumption, to a pair x, y of different points there is a p ∈ P such that
ε := 1
2
p(x − y) 0. Then U(x) := x + Bε(p) and U(y) := y + Bε(p) are disjoint
neighborhoods, so the axiom T2 is valid. The continuity of summation at the point
[0, 0] follows from the inequality p(x+y) ≤ p(x)+p(y); for the scalar multiplication
we use p(αx−α0x0) ≤ |α − α0| p(x0) + |α| p(x−x0).
A topological vector space with a topology induced by a family P separat-
ing points is called locally convex. This name has an obvious motivation: if x, y ∈
Bε(p1, . . . , pn), then pj(tx+(1−t)x) ≤ tpj(x)+(1−t)pj(x) holds for any t ∈ [0, 1]
so the sets Bε(p1, . . . , pn) are convex. The convexity is preserved at translations, so
the local basis of τP
at each x ∈ V consists of convex sets (see also the notes).
1.4.3 Example: The family P := {px := |(x, ·)| : x ∈ V } in a pre–Hilbert space V
generates a locally convex topology which is called the weak topology and is denoted
as τw; it is easy to see that it is weaker than the “natural” topology induced by the
norm.
1.4.4 Theorem: A locally convex space (V, τ) is metrizable iff there is a countable
family P of seminorms which generates the topology τ.
Proof: If V is metrizable it is first countable. Let {Uj : j = 1, 2, . . .} be a local
basis of τ at the point 0. By definition, to any Uj we can find ε 0 and a
finite subsystem Pj ⊂ P such that p∈Pj
Bε(p) ⊂ Uj. The family P
:= ∞
j=1 Pj
is countable and generates a topology τP
which is not stronger than τ := τP
;
the above inclusion shows that τP
= τ. On the other hand, suppose that τ is
generated by a family {pn : n = 1, 2, . . .} separating points; then we can define
a metric as in Problem 16 and show that the corresponding topology satisfies
τ = τ (Problem 43).](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-37-320.jpg)
![1.5 Banach spaces and operators on them 15
A locally convex space which is complete with respect to the metric used in the
proof is called a Fréchet space (see also the notes).
1.4.5 Example: The set S(Rn
) consists of all infinitely differentiable functions
f : Rn
→ C such that
f J,K := sup
x∈Rn
| xJ
(DK
f)(x)| ∞
holds for any multi–indices J := [j1, . . . , jn] , K := [k1, . . . , kn] with jr, kr non–
negative integers, where xJ
:= ξj1
1 . . . ξjn
n , DK
:= ∂|K|
/∂ξk1
1 . . . ∂ξkn
n and |K| :=
k1 + · · · + kn. It is easy to see that any such f and any derivative DK
f (as well
as polynomial combinations of them) tend to zero faster than |xJ
|−1
for each J;
we speak about rapidly decreasing functions. It is also clear that any · J,K is a
seminorm, with f 0,0 = f ∞, and the family P := { · J,K} separates points.
The corresponding locally convex space S(Rn
) is called the Schwartz space; one can
show that it is complete, i.e., a Fréchet space (see the notes).
An important subspace in S(Rn
) consists of infinitely differentiable functions
with a compact support; we denote it as C∞
0 (Rn
). It is dense,
C∞
0 (Rn) = S(Rn
) , (1.1)
with respect to the topology of S(Rn
) (Problem 44).
1.5 Banach spaces and operators on them
A normed space which is complete with respect to the norm–induced metrics is called
a Banach space. We have already met some frequently used Banach spaces —
see Example 1.2.1 and Problem 23. In view of Problem 21, any finite–dimensional
normed space is complete; in the general case we have the following completeness
criterion, the proof of which is left to the reader (see also Example 1.5.3b below).
1.5.1 Theorem: A normed space V is complete iff to any sequence {xn} ⊂ V
such that
∞
n=1 xn ∞ there is an x ∈ V such that x = limn→∞
n
k=1 xk (or
in short, iff any absolutely summable sequence is summable).
Given a noncomplete norm space, we can always extend it to a Banach space by
the standard completion procedure (Problem 46). A set M in a Banach space X
is called total if Mlin = X. Such a set is a basis if M is linearly independent and
dim X ∞, while an infinite–dimensional space can contain linearly independent
total sets, which are not Hamel bases of X (cf. the notes to Section 1.1).
1.5.2 Lemma: (a) If M is total in a Banach space X, then any set N ⊂ X dense
in M is total in X.
(b) A Banach space which contains a countable total set is separable.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-38-320.jpg)
![16 1 Some notions from functional analysis
Proof: Part (a) follows from the appropriate definitions. Suppose that M =
{x1, x2, . . .} is total in X and Crat is the countable set of complex numbers with ra-
tional real and imaginary parts; then the set L := {
n
j=1 γjxj : γj ∈ Crat, n ∞}
is countable. Since Crat is dense in C, we get L = X.
1.5.3 Examples: (a) The set P(a, b) of all complex polynomials on (a, b) is an
infinite–dimensional subspace in C[a, b] := C([a, b]). By the Weierstrass theo-
rem, any f ∈ C[a, b] can be approximated by a uniformly convergent sequence
of polynomials; hence C[a, b] is a complete envelope of (P(a, b), · ∞). The
set {xk
: k = 0, 1, . . . } is total in C[a, b], which is therefore separable.
(b) Consider the sequences Ek := {δjk}∞
j=1 in p
, p ≥ 1. For a given X := {ξj} ∈
p
, the sums Xn :=
n
j=1 ξjEj are nothing else than truncated sequences, so
limn→∞ X−Xn p = 0. Hence {Ek : k = 1, 2 . . .} is a countable total set and
p
is separable. Notice also that the sequence {ξjEj}∞
j=1 is summable but it
may not be absolutely summable for p 1.
(c) Consider next the space Lp
(Rn
, dµ) with an arbitrary Borel measure µ on
Rn
. We use the notation of Appendix A. In particular, J n
is the family of all
bounded intervals in Rn
; then we define S(n)
:= {χJ : J ∈ J n
}. It is a subset
in Lp
and the elements of its linear envelope are called step functions; we can
check that S(n)
is total in Lp
(Rn
, dµ) (Problem 47). Combining this result
with Lemma 1.5.2 we see that the subspace C∞
0 (Rn
) is dense in Lp
(Rn
, dµ); in
particular, for the Lebesgue measure on Rn
the inclusions C∞
0 (Rn
) ⊂ S(Rn
) ⊂
Lp
(Rn
) yield
(C∞
0 (Rn))p = (S(Rn))p = Lp
(Rn
) . (1.2)
(d) Given a topological space (X, τ) we call C∞(X) the set of all continuous
functions on X with the following property: for any ε 0 there is a compact
set K ⊂ X such that |f(x)| ε outside K. It is not difficult to check that
C∞(X) is a closed subspace in C(X) and C0(X) = C∞(X), where C0(X)
is the set of continuous functions with compact support (Problem 48). In the
particular case X = Rn
, C∞
0 (Rn
) is dense in C∞(Rn
) (see the notes), so
(C∞
0 (Rn))∞ = (S(Rn))∞ = C∞(Rn
) . (1.3)
There are various ways in, which it is possible to construct new Banach spaces
from given ones. We mention two of them (see also Problem 49):
(i) Let {Xj : j = 1, 2, . . .} be a countable family of Banach spaces. We denote
by X the set of all sequences x := {xj} , xj ∈ Xj, such that
j xj j ∞ ,
and equip it with the “componentwise” defined summation and scalar mul-
tiplication. The norm X ⊕ :=
j xj j turns it into a Banach space; the
completeness can be checked as for p
(Problem 23). The space (X, · ⊕) is
called the direct sum of the spaces Xj , j = 1, 2, . . ., and denoted as
⊕
j Xj.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-39-320.jpg)
![24 1 Some notions from functional analysis
1.6.3 Lemma: Let B ∈ B(X, Y) and ε 0. If (BUε)o
= ∅ or (BUε)o
= ∅; then
0 ∈ (BUη)0
or 0 ∈ (BUη)0
, respectively, holds for any η 0.
Proof: Let y0 be an interior point of BUε; then there is δ 0 such that y0 +
Vδ ⊂ BUε, i.e., to any y ∈ Vδ there exists a sequence {x
(y)
n } ⊂ Uε such that
z
(y)
n := Bx
(y)
n → y + y0. In particular, z
(0)
n → y0, so z
(y)
n − z
(0)
n → y, and since
x
(y)
n −x
(o)
n X 2ε we get Vδ ⊂ BU2ε. In view of Problem 33, cBUε = cBUε holds
for any c 0. This implies that Vη ⊂ BUη , η
:= ηδ
2ε
, so 0 is an interior point of
BUη. A similar argument applies to (BUε)o
= ∅.
Proof of Theorem 1.6.2: To any x ∈ G we can find Uη such that x + Uη ⊂ G, i.e.,
Bx+BUη ⊂ BG. If there is Vδ ⊂ BUη the set BG is open; hence it is sufficient to
check 0 ∈ (BUη)o
for any η 0. We write X = ∞
n=1 Un; since B is surjective, we
have Y = ∞
n=1 BUn and the Baire category theorem implies (BUñ)o
= ∅ for some
positive integer ñ. We shall prove that BUñ ⊂ BU2ñ.
Due to the lemma, BUñ contains a ball Vδ, and this further implies Vδj
⊂
BUnj
for j = 1, 2, . . ., where δj := δ/2j
and nj := ñ/2j
. Let y ∈ BUñ, so any
neighborhood of y contains elements of BUñ; in particular, for the neighborhood
y+Vδ1 we can find x1 ∈ Uñ such that Bx1 ∈ y+Vδ1 , and therefore also y−Bx1 ∈
Vδ1 ⊂ BUn1 . Repeating the argument we see that there is an x2 ∈ Un1 such that
y−Bx1 −Bx2 ∈ Vδ2 ⊂ BUn2 etc; in this way we construct a sequence {xj} ⊂ X
such that xj X 2ñ/2j
and
y −
j
k=1
Bxk
Y
δj .
Then Theorem 1.6.1 implies the existence of limj→∞
j
k=1 xj =: x ∈ X; we have
y = Bx because B is continuous. Now x X ≤
∞
k=1 xk X 2ñ, so y ∈ BU2ñ;
this proves BUñ ⊂ BU2ñ. Since (BUñ)o
= ∅ the set BU2ñ has an interior point;
using the lemma again we find 0 ∈ (BUη)o
for any η 0.
Theorem 1.6.2 further implies the following often used result, the proof of which
is left to the reader (Problem 58).
1.6.4 Corollary (inverse–mapping theorem): If B ∈ B(X, Y) is a bijection, then
B−1
is a continuous linear operator from Y to X.
In the rest of this section, T means a linear operator defined on a subspace DT
of a Banach space X and mapping DT into a Banach space Y; we do not assume
T to be continuous. The subspace DT is called the domain of the operator T and
is alternatively denoted as D(T).
The set Γ(T) := { [x, Tx] : x ∈ DT } is called the graph of the operator T ;
it is a subspace in the Banach space X ⊕ Y. In general, a subspace Γ ⊂ X ⊕ Y is
said to be a graph if each element [x, y] ∈ Γ is determined by its first component.
Any graph Γ determines the linear operator TΓ from X to Y with the domain
D(TΓ) := { x ∈ X : [x, y] ∈ Γ } by TΓx := y for each [x, y] ∈ Γ. It is clear that](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-47-320.jpg)
![1.7 Spectra of closed linear operators 25
Γ(TΓ) = Γ and conversely, a linear operator is uniquely determined by its graph,
TΓ(T) = T. An operator T is called closed if Γ(T) is a closed set in X ⊕ Y; the
definition of the direct product gives the following equivalent expression.
1.6.5 Proposition: An operator T is closed iff for any sequence {xn} ⊂ DT such
that xn → x and Txn → y, we have x ∈ DT and y = Tx.
Any bounded operator is obviously closed. On the other hand, there are closed
operators which are not bounded (Problem 59). If Γ(T) is not closed but its closure
in X ⊕ Y is a graph (which may not be true) the operator T is said to be closable
and the closed operator
T := TΓ(T)
is called the closure of the operator T; we have Γ(T) = Γ(T). Since Γ(T) is
the smallest closed set containing Γ(T), the closure is the smallest closed extension
of the operator T. Moreover, Γ(T) is a subspace in X ⊕ Y, so it is a graph iff
[0, y] ∈ Γ(T) implies y = 0. This property makes it possible to describe the closure
sequentially.
1.6.6 Proposition: (a) An operator T is closable iff any sequence {xn} ⊂ DT
such that xn → 0 and Txn → y fulfils y = 0.
(b) A vector x belongs to D(T) iff T is closable and there is a sequence
{xn} ⊂ DT such that xn → x and {Txn} is convergent; if this is true,
we have Txn → Tx.
We have mentioned that a closed operator may not be continuous. However, this
can happen only if the operator is not defined on the whole space X.
1.6.7 Theorem (closed–graph theorem): A closed linear operator T : X → Y with
DT = X is continuous.
Proof: Γ(T) is by assumption a closed subspace in X ⊕ Y, so it is a Banach space
with the direct–sum norm, [x, y] ⊕ = x X + y Y. The map S1 : S1([x, Tx]) = x
is a continuous linear bijection from Γ(T) to X, and therefore S−1
1 is continuous
due to Corollary 1.6.4. The map S2 : S2([x, Tx]) = Tx is again continuous, and the
same is true for the composite map S2◦S−1
1 = T.
1.7 Spectra of closed linear operators
We denote by C(X) the set of all closed linear operators from a Banach space X
to itself; since such operators are not necessarily bounded, one has to pay atten-
tion to their domains. A complex number λ is called an eigenvalue of an operator
T ∈ C(X) if T −λI is not injective, i.e., if there is a nonzero vector x ∈ DT
such that Tx = λx. Any vector with this property is called an eigenvector of T
(corresponding to the eigenvalue λ ). The subspace Ker (T −λ) is the respective](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-48-320.jpg)
![Notes to Chapter 1 29
1.7.6 Examples: (a) Consider the operators T and T(0)
:= T|
D(0)
of Problem 59.
The function fλ : fλ(x) = eλx
belongs to DT and Tfλ = λfλ, so σ(T) =
σp(T) = C and ρ(T) = ∅. On the other hand, for a given λ ∈ C we define
the operator Sλ : (Sλg)(x) =
x
0
eλ(x−t)
g(t) dt which maps C[0, 1] to D(0)
.
We have (Sλg)
− λ(Sλg) = g for any g ∈ C[0, 1] and Sλ(f
− λf) = f for all
f ∈ D(0)
, and therefore Sλ = (T(0)
−λ)−1
; in view of the closed–graph theorem
and Problem 60b, Sλ is bounded, so we get ρ(T(0)
) = C and σ(T(0)
) = ∅.
(b) The same conclusions are valid for the operator P̃ : P̃f := −if
on L2
(0, 1)
whose domain D(P̃) := AC[0, 1] consists of all functions that are absolutely
continuous on the interval [0, 1] with the derivatives in L2
(0, 1), and its re-
striction P(0)
:= P |
{f ∈ AC[0, 1] : f(0) = 0 }. However, one has to use a
different method to check that P̃ and P(0)
are closed; we postpone the proof
to Example 4.2.5.
Notes to Chapter 1
Section 1.1 A linearly independent set B ⊂ V such that Blin = V is called a Hamel
basis of the space V and its cardinality is the algebraic dimension of V . Such a basis
exists in any vector space (Problem 4). We are more interested, however, in other bases
which allow us to express elements of a space as “infinite linear combinations” of the basis
vectors. This requires a topology; we shall return to that problem in the next chapter.
A set C ⊂ V is convex if together with any two points it contains the line segment
connecting them. Any subspace L ⊂ V is convex, and the intersection of any family
of convex sets is again convex. If a point x ∈ C does not belong to the line segment
connecting some y, z ∈ C different from x, it is called extremal. Equivalently, x ∈ C is
an extremal point of C if x = ty + (1 − t)z with t ∈ (0, 1), y, z ∈ C implies x = y = z.
The Hahn–Banach theorem is proven in most functional–analysis textbooks — see,
e.g., [[ KF ]], [[ RS 1 ]], [[ Tay ]]. The set of all linear functionals on a given V becomes a
vector space if we equip it with the operations defined by (αf + g)(x) := αf(x) + g(x);
we call it the algebraic dual space of V and denote as V f . If V is finite–dimensional, one
can check easily that V and V f are isomorphic.
We often need to know whether a given family F of maps F : X → Y is “large
enough” to contain for any pair of different elements x, y ∈ X a map f such that
f(x) = f(y); if this is true we say that F separates points. If X, Y are vector spaces and
the maps in F are linear, the family separates points if for any non–zero x ∈ X there is
fx ∈ F such that fx(x) = 0. This is true, in particular, for F = Xf (Problem 7).
Section 1.2 As a by–product of the argument of Example 1.2.1 we have obtained the
following result: if a sequence {fn} ⊂ Lp(M, dµ) fulfils fn −f p → 0, then there is a
subsequence {fnk
} such that fnk
(x) → f(x) for µ–a.a. x ∈ M. In fact, this is true for
any positive measure and the space Lp(M, dµ) is still complete in this case — see, e.g.,
[[ Jar 2 ]], Thm.68; [[ DS ]], Sec.III.6.
An alternative way to introduce the notion of a topology is through axiomatization
of properties of the closure — see Problem 28 and [[ Kel ]] for more details. The described
construction of the topological product extends easily to any finite number of topological](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-52-320.jpg)
![30 1 Some notions from functional analysis
spaces; for further generalizations see, e.g., [[Nai 1]], Sec.I.2.12. A discussion of the count-
ability and separability axioms and their relations can be found in topology monographs
such as [[ Al ]] or [[ Kel ]], and also in most functional–analysis textbooks. For instance, any
second countable regular space is normal; the fundamental result of Uryson claims that
a second countable space is normal iff its topology is induced by a metric, thus giving a
partial answer to the metrizability problem.
A topological space (X, τ) is connected if it cannot be written as a union of two
non–empty disjoint open sets; this is equivalent to the requirement that X and ∅ are the
only two sets which are simultaneously closed and open. A set M ⊂ X is connected if the
space (M, τM ) with the induced topology is connected. A continuous map ϕ : [0, 1] → X
is called a curve connecting the points ϕ(0) and ϕ(1). A topological space (or its subset)
whose any two points can be connected by a curve is said to be arcwise (or linearly)
connected. Such spaces are connected but the converse is not true.
Section 1.3 There is a standard procedure called one–point compactification which allows
us to construct for a given noncompact (X, τ) a compact space (X, τ) such that (X, τ)
is its subspace and X X ≡ {x0} is a one–point set; the topology τ consists of all sets
{x0}∪(X F), where F is a closed compact set in X — for more details see, e.g., [[Tay]],
Sec.2.31. A simple example is the compactification of C by adding to it the point ∞.
Any compact space is by definition σ–compact but the converse is not true — cf. [[KF]],
Sec.IV.6.4. An infinite set in a σ–compact space has again at least one accumulation point;
this is clear from the proof of Proposition 1.3.1. In some cases the notions of compactness
and σ–compactness coincide (cf. Corollary 1.3.5 and Problem 37c).
A net {yβ}β∈J is called a subnet of a net {xα}α∈I if there is a map ϕ : J → I such
that (i) yβ = xϕ(β) for all β ∈ J, (ii) for any α ∈ J there is a β ∈ J such that β β
implies ϕ(β) α. Using this definition, we can state the Bolzano–Weierstrass theorem:
A topological space is compact iff any net in X has a convergent subnet. A proof can be
found, e.g., in [[ RS 1 ]], Sec.IV.3.
A map f of a metric space (X, ) to (X, ) is uniformly continuous if to any ε 0
there is δ 0 such that (f(x1), f(x2)) ε holds for any pair of points x1, x2 ∈ X
fulfilling (x1, x2) ε. One can easily prove the following useful result: a continuous map
of a compact space (X, ) to (X, ) is uniformly continuous.
Section 1.4 A converse to Theorem 1.4.1 was proved by F. Riesz: any locally compact
topological vector space is finite–dimensional — see, e.g., [[ Tay ]], Sec.3.3. Some metric–
space notions do not extend directly to topological spaces but can be used after a suitable
generalization. For instance, a set M in a topological vector space is bounded if to any
neighborhood U of zero there is an α ∈ C such that M ⊂ αU. In view of Problem 42b,
there is a positive b such that M ⊂ βU holds for all |β| ≥ b. It is easy to see that in
a normed space this definition is equivalent to the requirement of existence of a cM 0
such that x cM for all x ∈ M.
A locally convex space is often defined as a topological vector space in which any
neighborhood of the point 0 contains a convex neighborhood of zero. We have seen that
this is true for (V, τP). On the other hand, if a topological vector space (V, τ) has the
stated property, there is a family of seminorms on V which separate points and generate
the topology τ — see, e.g., [[ Tay ]], Sec.3.8 — so the two definitions are really equivalent.
A net {xα}α∈I in a topological vector space V is Cauchy if for any neighborhood U
of 0 there is a γ ∈ I such that xα−xβ ∈ U holds for all α γ , β γ. In particular, in](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-53-320.jpg)
![Notes to Chapter 1 31
a locally convex space with a topology generated by a family P the condition reads: for
any ε 0 and p ∈ P, there is a γ ∈ I such that p(xα −xβ) ε for all α γ , β γ.
The space V is complete if any Cauchy net in it converges to some x ∈ V . In a similar
way, we define a Cauchy net in a metric space. Since such a net in a complete metric space
is convergent — see, e.g., [[ DS 1 ]], Lemma I.7.5 — a locally convex space with a topology
generated by a countable family P is Fréchet iff it is complete in the sense of the above
definition. The completeness of S(Rn) is proved, e.g., in [[ RS 1 ]], Sec.V.3.
Section 1.5 Combining the results of Example 1.5.3c with the inclusion C∞
0 (Rn) ⊂
C(Rn) ∩ Lp(Rn, dµ) we find that the set C(Rn) ∩ Lp(Rn, dµ) is also dense in Lp(Rn, dµ).
This can be proved directly for a wider class of Lp spaces — cf. [[ KF ]], Sec.VII.1.2. The
proof of density of C∞
0 (Rn) in C∞(Rn) can be found, e.g., in [[ Yo ]], Sec.1.1. In a similar
way, the set C∞
0 (Ω) for an open connected set Ω ⊂ Rn, which we shall mention below, is
dense in L2(Ω) — cf. [[ RS 4 ]], Sec.XIII.14.
The map F : L1(Rn) → C∞(Rn) is injective as can checked easily: the relation
Fg = 0 for g ∈ L1 implies
Rn f(x)g(x) dx = 0 for all f ∈ S(Rn); then
J g(x) dx = 0
holds by Problem 47 for any bounded interval J ⊂ Rn and therefore also for any Borel
set in Rn; this in turn means g = 0. On the other hand, F is not surjective — see, e.g.,
[[ KF ]], Sec.VIII.4.7; [[ Jar 2 ]], Sec.XIII.11.
The product of functions f, g ∈ S(Rn) belongs to S(Rn). The relation from Problem 52
implies that its Fourier transform is
fg = (2π)−n/2 ˆ
f ∗ ĝ, where
(f ∗ g)(x) :=
Rn
f(x−y)g(y) dy =
Rn
g(x−y)f(y) dy .
The map [f, g] → f ∗ g is called the convolution. It is a binary operation on S(Rn) which
is obviously bilinear and commutative; the relation f ∗g = (2π)n/2
ˇ
fǧ shows that it is also
associative. Some important extensions of the convolution are discussed in [[Yo]], Sec.VI.3;
[[ RS 2 ]], Sec.IX.1.
While ( 1)∗ is isomorphic to ∞, the dual ( ∞)∗ is not isomorphic to 1. This follows
from Corollary 1.5.8c because 1 is separable but ∞ is nonseparable due to Problem 18.
The situation with the spaces Lp(X, dµ) is similar: (Lp)∗ , p ≥ 1, is linearly isomorphic
to Lq by fϕ : fϕ(ψ) =
X ϕψ dµ — see [[ Ru 1 ]], Sec.6.16; [[ Tay ]], Sec.7.4. The spaces
(L∞)∗ and L1 are again nonisomorphic with the exception of the trivial case when the
measure µ has a finite support; the expression of (L∞)∗ is given in [[ DS 1 ]], Sec.IV.8.
These results have implications for the reflexivity of the considered spaces as mentioned
in the text. The proof of Theorem 1.5.10 can be found in [[ DS 1 ]], Sec.II.3.23.
Another example of Banach spaces in which one can find a general form of a bounded
linear functional is represented by the spaces C(K) and CR(K) of continuous functions
(complex or real–valued, respectively) on a compact Hausdorff space K. The Riesz–Markov
theorem associates the functionals with Borel measures on K. In the particular case of
X ≡ CR[a, b] any such measure corresponds to a function F of a bounded variation; the
theorem then claims that for any f ∈ X∗ there is F of a bounded variation such that
f(ϕ) =
[a,b] ϕ dF holds for all ϕ ∈ X, and moreover, f is equal to the total variation
of F — for details see, e.g., [[ KF ]], Sec.VI.6.6 or [[ RN ]], Sec.50. The general formulation
and proof of the Riesz–Markov theorem can be found in [[ DS 1 ]], Sec.IV.6.
One of the most important examples of duals to topological vector spaces is the space
S(Rn) whose elements are called tempered distributions, and the dual of the complete](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-54-320.jpg)
![Mrs. de Trappe. You seem absent-minded, my dear Edith. [Pause.] I
must be going now. Where are Arthur and James? We have not a
moment to lose. We are going to choose wedding presents. James is
going to choose Arthur's and Arthur is going to choose James's, so
there can be no jealousy. It was I who thought of that way out of the
difficulty. One does one's best to be nice to them, and then
something happens and upsets all one's plans. Where is Cyril?
Lady Dol. I am afraid Cyril is not at home.
Mrs. de Trappe. Then I shall not see him. Tell him I am angry, and
give my love to Julia. I hope she does not disturb you when you are
in the drawing-room and have visitors. So difficult to keep a grown-
up girl out of the drawing-room. Where can those men be? [Enter
Lord Doldrummond, Mr. Featherleigh, and Mr. Banish.] Ah! here they
are. Now, come along; we haven't a moment to lose. Good-bye,
Edith.
[Exeunt (after wishing their adieux) Mrs. de Trappe, Mr. Featherleigh,
and Mr. Banish, Lord Doldrummond following them.]
Lady Dol. [Stands alone in the middle of the room, repeating.] Cyril
and—Sarah Sparrow! My son and Sarah Sparrow! And he has met
her through the one woman for whom I have been wrong enough to
forget my prejudices. What a punishment!
[Julia enters cautiously. She is so unusually beautiful that she barely
escapes the terrible charge of sublimity. But there is a certain
peevishness in her expression which adds a comfortable
smack of human nature to her classic features.]
Julia. I thought mamma would never go. I have been hiding in your
boudoir ever since I heard she was here.
Lady Dol. Was Cyril with you?
Julia. Oh, no; he has gone out for a walk.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-56-320.jpg)
![Lady Dol. Tell me, dearest, have you and Cyril had any disagreement
lately? Is there any misunderstanding?
Julia. Oh, no. [Sighs.]
Lady Dol. I remember quite well that before I married Herbert he
often suffered from the oddest moods of depression. Several times
he entreated me to break off the engagement. His affection was so
reverential that he feared he was not worthy of me. I assure you I
had the greatest difficulty in overcoming his scruples, and persuading
him that whatever his faults were I could help him to subdue them.
Julia. But Cyril and I are not engaged. It is all so uncertain, so
humiliating.
Lady Dol. Men take these things for granted. If the truth were
known, I daresay he already regards you as his wife.
Julia. [With an inspired air.] Perhaps that is why he treats me so
unkindly. I have often thought that if he were my husband he could
not be more disagreeable! He has not a word for me when I speak to
him. He does not hear. Oh, Lady Doldrummond, I know what is the
matter. He is in love, but I am not the one. You are all wrong.
Lady Dol. No, no, no. He loves you; I am sure of it. Only be patient
with him and it will come all right. Hush! is that his step? Stay here,
darling, and I will go into my room and write letters. [Exit, brushing
the tears from her eyes.]
[Butler ushers in Mr. Mandeville. Neither of them perceive Julia, who
has gone to the window.]
Butler. His Lordship will be down in half an hour, sir. He is now having
his hair brushed.
Julia. [In surprise as she looks round.] Mr. Mandeville! [Pause.] I
hardly expected to meet you here.
Mandeville. And why, may I ask?](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-57-320.jpg)
![Julia. You know what Lady Doldrummond is. How did you overcome
her scruples?
Mandeville. Is my reputation then so very bad?
Julia. You—you are supposed to be rather dangerous. You sing on
the stage, and have a tenor voice.
Mandeville. Is that enough to make a man dangerous?
Julia. How can I tell? But mamma said you were invincible. You
admire mamma, of course. [Sighs.]
Mandeville. A charming woman, Mrs. de Trappe. A very interesting
woman; so sympathetic.
Julia. But she said she would not listen to you
.Mandeville. Did she say that? [A slight pause.] I hope you will not be
angry when I own that I do not especially admire your mother. A
quarter of a century ago she may have had considerable attractions,
but—are you offended?
Julia. Offended? Oh, no. Only it seems strange. I thought that all
men admired mamma. [Pause.] You have not told me yet how you
made Lady Doldrummond's acquaintance.
Mandeville. I am here at Lord Aprile's invitation. He has decided that
he feels no further need of Lady Doldrummond's apron-strings.
Julia. Oh, Mr. Mandeville, are you teaching him to be wicked?
Mandeville. But you will agree with me that a young man cannot
make his mother a kind of scribbling diary?
Julia. Still, if he spends his time well, there does not seem to be any
reason why he should refuse to say where he dines when he is not at
home.
Mandeville. Lady Doldrummond holds such peculiar ideas; she would
find immorality in a sofa-cushion. If she were to know that Cyril is](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-58-320.jpg)
![coming with me to the dress rehearsal of our new piece!
Julia. It would break her heart. And Lord Doldrummond would be
indignant. Mamma says his own morals are so excellent!
Mandeville. Is he an invalid?
Julia. Certainly not. Why do you ask?
Mandeville. Whenever I hear of a charming husband I always think
that he must be an invalid. But as for morals, there can be no harm
in taking Cyril to a dress rehearsal. If you do not wish him to go,
however, I can easily say that the manager does not care to have
strangers present. [Pause.] Afterwards there is to be a ball at Miss
Sparrow's.
Julia. Is Cyril going there, too?
Mandeville. I believe that he has an invitation, but I will persuade
him to refuse it, if you would prefer him to remain at home.
Julia. You are very kind, Mr. Mandeville, but it is a matter of
indifference to me where Lord Aprile goes.
Mandeville. Perhaps I ought not to have mentioned this to you?
Julia. [Annoyed.] It does not make the least difference. In fact, I am
delighted to think that you are taking Cyril out into the world. He is
wretched in this house. [With heroism.] I am glad to think that he
knows anyone so interesting and clever and beautiful as Sarah
Sparrow. I suppose she would be considered beautiful?
Mandeville. [With a profound glance.] One can forget her—
sometimes.
Julia. [Looking down.] Perhaps—when I am as old as she is—I shall
be prettier than I am at present.
Mandeville. You always said you liked my voice. We never see
anything of each other now. I once thought that—well—that you](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-59-320.jpg)
![might like me better. Are you sure you are not angry with me
because I am taking Cyril to this rehearsal?
Julia. Quite sure. Why should I care where Cyril goes? I only wish
that I, too, might go to the theatre to-night. What part do you play?
And what do you sing? A serenade?
Mandeville. [Astounded.] Yes. How on earth did you guess that? The
costume is, of course, picturesque, and that is the great thing in an
opera. A few men can sing—after a fashion—but to find the right
clothes to sing in—that shows the true artist.
Julia. And Sarah; does she look her part?
Mandeville. Well, I do not like to say anything against her, but she is
not quite the person I should cast for la Marquise de la Perdrigonde.
Ah! if you were on the stage, Miss de Trappe! You have just the
exquisite charm, the grace, the majesty of bearing which, in the
opinion of those who have never been to Court, is the peculiar
distinction of women accustomed to the highest society.
Julia. Oh, I should like to be an actress!
Mandeville. No! no! I spoke selfishly—if you only acted with me, it
would be different; but—but I could not bear to see another man
making love to you—another man holding your hand and singing into
your eyes—and—and——Oh, this is madness. You must not listen to
me.
Julia. I am not—angry, but—you must never again say things which
you do not mean. If I thought you were untruthful it would make me
so—so miserable. Always tell me the truth. [Holds out her hand.]
Mandeville. You are very beautiful!
[She drops her eyes, smiles, and wanders unconsciously to the
mirror.]](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-60-320.jpg)
![[Lady Doldrummond suddenly enters from the boudoir, and Cyril
from the middle door. Cyril is handsome, but his features have
that delicacy and his expression that pensiveness which
promise artistic longings and domestic disappointment.]
Cyril. [Cordially and in a state of suppressed excitement.] Oh,
mother, this is my friend Mandeville. You have heard me mention
him?
Lady Dol. I do not remember, but——
Cyril. When I promised to go out with you this afternoon, I forgot
that I had another engagement. Mandeville has been kind enough to
call for me.
Lady Dol. Another engagement, Cyril?
[Lord Doldrummond enters and comes down, anxiously looking from
one to the other.]
Cyril. Father, this is my friend Mandeville. We have arranged to go up
to town this afternoon.
Lady Dol. [Calmly.] What time shall I send the carriage to the station
for you? The last train usually arrives about——
Cyril. I shall not return to-night. I intend to stay in town. Mandeville
will put me up.
Lord Dol. And where are you going?
Mandeville. He is coming to our dress rehearsal of the “Dandy and
the Dancer.”
Cyril. At the Parnassus. [Lord and Lady Doldrummond exchange
horrified glances.] I daresay you have never heard of the place, but it
amuses me to go there, and I must learn life for myself. I am two-
and-twenty, and it is not extraordinary that I should wish to be my
own master. I intend to have chambers of my own in town.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-61-320.jpg)
![Lady Dol. Surely you have every liberty in this house?
Lord Dol. If you leave us, you will leave the rooms in which your
mother has spent every hour of her life, since the day you were
born, planning and improving. Must all her care and thought go for
nothing? The silk hangings in your bedroom she worked with her
own hands. There is not so much as a pen-wiper in your quarter of
the house which she did not choose with the idea of giving you one
more token of her affection.
Cyril. I am not ungrateful, but I cannot see much of the world
through my mother's embroidery. As you say, I have every comfort
here. I may gorge at your expense and snore on your pillows and
bully your servants. I can do everything, in fact, but live. Dear
mother, be reasonable. [Tries to kiss her. She remains quite frigid.]
[Footman enters.]
Footman. The dog-cart is at the door, my lord.
Cyril. You think it well over and you will see that I am perfectly right.
Come on, Mandeville, we shall miss the train. Make haste: there is no
time to be polite. [He goes out, dragging Mandeville after him, and
ignoring Julia.]
Lord Dol. Was that my son? I am ashamed of him! To desert us in
this rude, insolent, heartless manner. If I had whipped him more and
loved him less, he would not have been leaving me to lodge with a
God knows who. I disown him! The fool!
Lady Dol. If you have anything to say, blame me! Cyril has the
noblest heart in the world; I am the fool.
Curtain.](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-62-320.jpg)
![[The following text was cropped out of the
illustration:
Boston: Copeland Day
Price 5/-]
Inner Cover: The Yellow Book
An Illustrated Quarterly
Volume I April 1894
London: Elkin Mathews John Lane
Boston: Copeland Day](https://image.slidesharecdn.com/4985933-250526112841-90da6548/85/Hilbert-Space-Operators-In-Quantum-Physics-2ed-Blank-Jir-Exner-64-320.jpg)
Hilbert Space Operators In Quantum Physics 2ed Blank Jir Exner
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- 6. Theoretical and Mathematical Physics The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary back- ground materials, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research. Editorial Board W. Beiglböck, Institute of Applied Mathematics, University of Heidelberg, Germany J.-P. Eckmann, Department of Theoretical Physics, University of Geneva, Switzerland H. Grosse, Institute of Theoretical Physics, University of Vienna, Austria M. Loss, School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA S. Smirnov, Mathematics Section, University of Geneva, Switzerland L. Takhtajan, Department of Mathematics, Stony Brook University, NY, USA J. Yngvason, Institute of Theoretical Physics, University of Vienna, Austria
- 8. Jiří Blank • Pavel Exner • Miloslav Havlíček Hilbert Space Operators in Quantum Physics Second Edition ABC
- 9. Jiří Blank† Prague Czechia Pavel Exner Doppler Institute Břehová 7 11519 Prague and Nuclear Physics Institute Czech Academy of Sciences 25068 Řež near Prague Czech Republic exner@ujf.cas.cz Miloslav Havlíček Doppler Institute and Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Trojanova 13 12000 Prague Czech Republic havlicek@fjfi.cvut.cz ISBN 978-1-4020-8869-8 e-ISBN 978-1-4020-8870-4 Library of Congress Control Number: 2008933703 All Rights Reserved c 2008 Springer Science + Business Media B.V. c 1993, first edition, AIP, Melville, NY No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Charles University
- 10. To our wives and daughters
- 12. Preface to the second edition Almost fifteen years later, and there is little change in our motivation. Mathemat- ical physics of quantum systems remains a lively subject of intrinsic interest with numerous applications, both actual and potential. In the preface to the first edition we have described the origin of this book rooted at the beginning in a course of lectures. With this fact in mind, we were naturally pleased to learn that the volume was used as a course text in many points of the world and we gladly accepted the offer of Springer Verlag which inherited the rights from our original publisher, to consider preparation of a second edition. It was our ambition to bring the reader close to the places where real life dwells, and therefore this edition had to be more than a corrected printing. The field is developing rapidly and since the first edition various new subjects have appeared; as a couple of examples let us mention quantum computing or the major progress in the investigation of random Schrödinger operators. There are, however, good sources in the literature where the reader can learn about these and other new developments. We decided instead to amend the book with results about new topics which are less well covered, and the same time, closer to the research interests of one of us. The main change here are two new chapters devoted to quantum waveguides and quantum graphs. Following the spirit of this book we have not aspired to full coverage — each of these subjects would deserve a separate monograph — but we have given a detailed enough exposition to allow the interested reader to follow (and enjoy) fresh research results in this area. In connection with this we have updated the list of references, not only in the added chapters but also in other parts of the text in the second part of the book where we found it appropriate. Naturally we have corrected misprints and minor inconsistencies spotted in the first edition. We thank the colleagues who brought them to our attention, in particu- lar to Jana Stará, who indicated numerous improvements. As with the first edition, we have asked a native speaker to try to remove the foreign “accent” from our writing; we are grateful to Mark Harmer for accepting this role. Prague, December 2007 Pavel Exner Miloslav Havlı́ček vii
- 14. Preface Relations between mathematics and physics have a long and entangled tradition. In spite of repeated clashes resulting from the different aims and methods of the two disciplines, both sides have always benefitted. The place where contacts are most intensive is usually called mathematical physics, or if you prefer, physical mathematics. These terms express the fact that mathematical methods are needed here more to understand the essence of problems than as a computational tool, and conversely, the investigated properties of physical systems are also inspiring from the mathematical point of view. In fact, this field does not need any advocacy. When A. Wightman remarked a few years ago that it had become “socially acceptable”, it was a pleasant under- statement; by all accounts, mathematical physics is flourishing. It has long left the adolescent stage when it cherished only oscillating strings and membranes; nowadays it has built synapses to almost every part of physics. Evidence that the discipline is developing actively is provided by the fruitful oscillation between the investigation of particular systems and synthetizing generalizations, as well as by discoveries of new connections between different branches. The drawback of this rapid development is that it has become virtually impos- sible to write a textbook on mathematical physics as a single topic. There are, of course, books which cover a wide range of problems, some of them indeed monu- mental, but even they are like cities which govern the territory while watching the frontier slowly moving towards the gray distance. This is simply the price we have to pay for the flood of ideas, concepts, tools, and results that our science is producing. It was not our aim to write a poor man’s version of some of the big textbooks. What we want is to give students basic information about the field, by which we mean an amount of knowledge that could constitute the basis of an intensive one– year course for those who already have the necessary training in algebra and analysis, as well as in classical and quantum mechanics. If our exposition should kindle interest in the subject, the student will be able, after taking such a course, to read specialized monographs and research papers, and to discover a research topic to his or her taste. We have mentioned that the span of the contemporary mathematical physics is vast; nevertheless the cornerstone remains where it was laid by J. von Neumann, H. Weyl, and the other founding fathers, namely in regions connected with quantum theory. Apart from its importance for fundamental problems such as the constitution of matter, this claim is supported by the fact that quantum theory is gradually ix
- 15. x Preface becoming a basis for most branches of applied physics, and has in this way entered our everyday life. The mathematical backbone of quantum physics is provided by the theory of linear operators on Hilbert spaces, which we discuss in the first half of this book. Here we follow a well–trodden path; this is why references in this part aim mostly at standard book sources, even for the few problems which maybe go beyond the stan- dard curriculum. To make the exposition self–contained without burdening the main text, we have collected the necessary information about measure theory, integration, and some algebraic notions in the appendices. The physical chapters in the second half are not intended to provide a self– contained exposition of quantum theory. As we have remarked, we suppose that the reader has background knowledge up to the level of a standard quantum mechan- ics course; the present text should rather provide new insights and help to reach a deeper understanding. However, we attempt to describe the mathematical founda- tions of quantum theory in a sufficiently complete way, so that a student coming from mathematics can start his or her way into this part of physics through our book. In connection with the intended purpose of the text, the character of referencing changes in the second part. Though the material discussed here is with a few excep- tions again standard, we try in the notes to each chapter to explain extensions of the discussed results and their relations to other problems; occasionally we have set traps for the reader’s curiosity. The notes are accompanied by a selective but quite broad list of references, which map ways to the areas where real life dwells. Each chapter is accompanied by a list of problems. Solving at least some of them in full detail is the safest way for the reader to check that he or she has indeed mastered the topic. The problem level ranges from elementary exercises to fairly complicated proofs and computations. We have refrained from marking the more difficult ones with asterisks because such a classification is always subjective, and after all, in real life you also often do not know in advance whether it will take you an hour or half a year to deal with a given problem. Let us add a few words about the history of the book. It originates from courses of lectures we have given in different forms during the past two decades at Charles University and the Czech Technical University in Prague. In the 1970s we prepared several volumes of lecture notes; ten years later we returned to them and rewrote the material into a textbook, again in Czech. It was prepared for publication in 1989, but the economic turmoil which inevitably accompanied the welcome changes delayed its publication, so that it appeared only recently. In the meantime we suffered a heavy blow. Our friend and coauthor, Jiřı́ Blank, died in February 1990 at the age of 50. His departure reminded us of the bitter truth that we usually are able to appreciate the real value of our relationships with fellow humans only after we have lost them. He was always a stabilizing element of our triumvirate of authors, and his spirit as a devoted and precise teacher is felt throughout this book; we hope that in this indirect way his classes will continue. Preparing the English edition was therefore left to the remaining two authors. It has been modified in many places. First of all, we have included two chapters and
- 16. Preface xi some other material which was prepared for the Czech version but then left out due to editorial restrictions. Though the aim of the book is not to report on the present state of research, as we have already remarked, the original manuscript was finished four years ago and we felt it was necessary to update the text and references in some places. On the other hand, since the audience addressed by the English text is different — and is equipped with different libraries — we decided to rewrite certain parts from the first half of the book in a more condensed form. One consequence of these alterations was that we chose to do the translation ourselves. This decision contained an obvious danger. If you write in a language which you did not master during your childhood, the result will necessarily contain some unwanted comical twists reminiscent of the famous character of Leo Rosten. We are indebted to P. Moylan and, in particular, to R. Healey, who have read the text and counteracted our numerous petty attacks against the English language; those clumsy expressions that remain are, of course, our own. There are many more people who deserve our thanks: coauthors of our research papers, colleagues with whom we have had the pleasure of exchanging ideas, and simply friends who have supported us during difficult times. We should not forget about students in our courses who have helped just by asking questions; some of them have now become our colleagues. In view of the book complex history, the list should be very long. We prefer to thank all of them anonymously. However, since every rule should have an exception, let us name J. Dittrich, who read the manuscript and corrected numerous mistakes. Last but not least we want to thank our wives, whose patience and understanding made the writing of this book possible. Prague, July 1993 Pavel Exner Miloslav Havlı́ček
- 18. Contents Preface to the second edition vii Preface ix 1 Some notions from functional analysis 1 1.1 Vector and normed spaces . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Metric and topological spaces . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Banach spaces and operators on them . . . . . . . . . . . . . . . . . . 15 1.6 The principle of uniform boundedness . . . . . . . . . . . . . . . . . . 23 1.7 Spectra of closed linear operators . . . . . . . . . . . . . . . . . . . . 25 Notes to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Hilbert spaces 41 2.1 The geometry of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 41 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Direct sums of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . 50 2.4 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Notes to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Bounded operators 63 3.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Hermitean operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Unitary and isometric operators . . . . . . . . . . . . . . . . . . . . . 72 3.4 Spectra of bounded normal operators . . . . . . . . . . . . . . . . . . 74 3.5 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6 Hilbert–Schmidt and trace–class operators . . . . . . . . . . . . . . . 81 Notes to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xiii
- 19. xiv Contents 4 Unbounded operators 93 4.1 The adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Closed operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3 Normal operators. Self–adjointness . . . . . . . . . . . . . . . . . . . 100 4.4 Reducibility. Unitary equivalence . . . . . . . . . . . . . . . . . . . . 105 4.5 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.7 Self–adjoint extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.8 Ordinary differential operators . . . . . . . . . . . . . . . . . . . . . . 126 4.9 Self–adjoint extensions of differential operators . . . . . . . . . . . . . 133 Notes to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5 Spectral theory 151 5.1 Projection–valued measures . . . . . . . . . . . . . . . . . . . . . . . 151 5.2 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4 Spectra of self–adjoint operators . . . . . . . . . . . . . . . . . . . . . 171 5.5 Functions of self–adjoint operators . . . . . . . . . . . . . . . . . . . . 176 5.6 Analytic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.7 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.8 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.9 Groups of unitary operators . . . . . . . . . . . . . . . . . . . . . . . 191 Notes to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 Operator sets and algebras 205 6.1 C∗ –algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.2 GNS construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.3 W∗ –algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.4 Normal states on W∗ –algebras . . . . . . . . . . . . . . . . . . . . . . 221 6.5 Commutative symmetric operator sets . . . . . . . . . . . . . . . . . 227 6.6 Complete sets of commuting operators . . . . . . . . . . . . . . . . . 232 6.7 Irreducibility. Functions of non-commuting operators . . . . . . . . . 235 6.8 Algebras of unbounded operators . . . . . . . . . . . . . . . . . . . . 239 Notes to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7 States and observables 251 7.1 Basic postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.2 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.3 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.4 Superselection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.5 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
- 20. Contents xv 7.6 The algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Notes to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8 Position and momentum 293 8.1 Uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.2 The canonical commutation relations . . . . . . . . . . . . . . . . . . 299 8.3 The classical limit and quantization . . . . . . . . . . . . . . . . . . . 306 Notes to Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9 Time evolution 317 9.1 The fundamental postulate . . . . . . . . . . . . . . . . . . . . . . . . 317 9.2 Pictures of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 9.3 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.4 The Feynman integral . . . . . . . . . . . . . . . . . . . . . . . . . . 330 9.5 Nonconservative systems . . . . . . . . . . . . . . . . . . . . . . . . . 334 9.6 Unstable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Notes to Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 10 Symmetries of quantum systems 357 10.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 10.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 10.3 General space–time transformations . . . . . . . . . . . . . . . . . . . 370 Notes to Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 11 Composite systems 379 11.1 States and observables . . . . . . . . . . . . . . . . . . . . . . . . . . 379 11.2 Reduced states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 11.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 11.4 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 11.5 Separation of variables. Symmetries . . . . . . . . . . . . . . . . . . . 392 Notes to Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 12 The second quantization 403 12.1 Fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 12.2 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . 408 12.3 Systems of noninteracting particles . . . . . . . . . . . . . . . . . . . 413 Notes to Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
- 21. xvi Contents 13 Axiomatization of quantum theory 425 13.1 Lattices of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . 425 13.2 States on proposition systems . . . . . . . . . . . . . . . . . . . . . . 430 13.3 Axioms for quantum field theory . . . . . . . . . . . . . . . . . . . . 434 Notes to Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 14 Schrödinger operators 443 14.1 Self–adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 14.2 The minimax principle. Analytic perturbations . . . . . . . . . . . . . 448 14.3 The discrete spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 454 14.4 The essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 462 14.5 Constrained motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 14.6 Point and contact interactions . . . . . . . . . . . . . . . . . . . . . . 474 Notes to Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 15 Scattering theory 491 15.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 15.2 Existence of wave operators . . . . . . . . . . . . . . . . . . . . . . . 498 15.3 Potential scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 15.4 A model of two–channel scattering . . . . . . . . . . . . . . . . . . . 510 Notes to Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 16 Quantum waveguides 527 16.1 Geometric effects in Dirichlet stripes . . . . . . . . . . . . . . . . . . 527 16.2 Point perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 16.3 Curved quantum layers . . . . . . . . . . . . . . . . . . . . . . . . . . 539 16.4 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Notes to Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 17 Quantum graphs 561 17.1 Admissible Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . 561 17.2 Meaning of the vertex coupling . . . . . . . . . . . . . . . . . . . . . 566 17.3 Spectral and scattering properties . . . . . . . . . . . . . . . . . . . . 570 17.4 Generalized graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 17.5 Leaky graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 Notes to Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
- 22. Contents xvii A Measure and integration 595 A.1 Sets, mappings, relations . . . . . . . . . . . . . . . . . . . . . . . . . 595 A.2 Measures and measurable functions . . . . . . . . . . . . . . . . . . . 598 A.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 A.4 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 A.5 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 B Some algebraic notions 609 B.1 Involutive algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 B.2 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 B.3 Lie algebras and Lie groups . . . . . . . . . . . . . . . . . . . . . . . 614 References 617 List of symbols 647 Index 651
- 24. Chapter 1 Some notions from functional analysis 1.1 Vector and normed spaces The notion of a vector space is obtained by axiomatization of the properties of the three–dimensional space of Euclidean geometry, or of configuration spaces of classical mechanics. A vector (or linear) space V is a set {x, y, . . . } equipped with the operations of summation, [x, y] → x + y ∈ V , and multiplication by a complex or real number α, [α, x] → αx ∈ V , such that (i) The summation is commutative, x + y = y + x, and associative, (x + y) + z = x+(y +z). There exist a zero element 0 ∈ V , and an inverse element −x ∈ V, to any x ∈ V so that x + 0 = x and x + (−x) = 0 holds for all x ∈ V . (ii) α(βx) = (αβ)x and 1x = x. (iii) The summation and multiplication are distributive, α(x + y) = αx + αy and (α + β)x = αx + βx. The elements of V are called vectors. The set of numbers (or scalars) in the definition can be replaced by any algebraic field F. Then we speak about a vector space over F, and in particular, about a complex and real vector space for F = C, R, respectively. A vector space without further specification in this book always means a complex vector space. 1.1.1 Examples: (a) The space Cn consists of n–tuples of complex numbers with the summation and scalar multiplication defined componentwise. In the same way, we define the real space Rn . (b) The space p , 1 ≤ p ≤ ∞, of all complex sequences X := {ξj}∞ j=1 such that ∞ j=1 |ξj|p ∞ for p ∞ and supj |ξj| ∞ if p = ∞, with the summation and scalar multiplication defined as above; the Minkowski inequality implies X + Y ∈ p for X, Y ∈ p (Problem 2). 1
- 25. 2 1 Some notions from functional analysis (c) The space C(J) of continuous complex functions on a closed interval J ⊂ R with (αf +g)(x) := αf(x)+g(x). In a similar way, we define the space C(X) of continuous functions on a compact X and spaces of bounded continuous functions on more general topological spaces (see the next two sections). A subspace L ⊂ V is a subset, which is itself a vector space with the same operations. A minimal subspace containing a given subset M ⊂ V is called the linear hull (envelope) of M and denoted as Mlin or lin(M). Vectors x1, . . . , xn ∈ V are linearly independent if α1x1 + · · · + αnxn = 0 implies that all the numbers α1, . . . , αn are zero; otherwise they are linearly dependent, which means some of them can be expressed as a linear combination of the others. A set M ⊂ V is linearly independent if each of its finite subsets consists of linearly independent vectors. This allows us to introduce the dimension of a vector space V as a maxi- mum number of linearly independent vectors in V . Among the spaces mentioned in Example 1.1.1, Cn and Rn are n–dimensional (Cn is 2n–dimensional as a real vec- tor space) while the others are infinite–dimensional. A basis of a finite–dimensional V is any linearly independent set B ⊂ V such that Blin = V ; it is clear that dim V = n iff V has a basis of n elements. Vector spaces V, V are said to be (algebraically) isomorphic if there is a bijection f : V → V , which is linear, f(αx + y) = αf(x) + f(y). Isomorphic spaces have the same dimension; for finite– dimensional spaces the converse is also true (Problem 3). There are various ways to construct new vector spaces from given ones. Let us mention two of them: (i) If V1, . . . , VN are vector spaces over the same field; then we can equip the Cartesian product V := V1 × · · · × VN with a summation and scalar multipli- cation defined by α[x1, . . . , xN ]+[y1, . . . , yN ] := [αx1 +y1, . . . , αxN +yN ]. The axioms are obviously satisfied; the resulting vector space is called the direct sum of V1, . . . , VN and denoted as V1 ⊕ · · · ⊕ VN or ⊕ j Vj. The same term and symbols are used if V1, . . . , VN are subspaces of a given space V such that each x ∈ V has a unique decomposition x = x1 + · · · + xN , xj ∈ Vj. (ii) If W is a subspace of a vector space V , we can introduce an equivalence relation on V by x ∼ y if x−y ∈ W. Defining the vector–space operations on the set Ṽ of equivalence classes by αx̃+ỹ := (αx+y)˜ for some x ∈ x̃, y ∈ ỹ, we get a vector space, which is called the factor space of V with respect to W and denoted as V/W. 1.1.2 Example: The space Lp (M, dµ) , p ≥ 1, where µ is a non–negative measure, consists of all measurable functions f : M → C satisfying M |f|p dµ ∞ with pointwise summation and scalar multiplication — cf. Appendix A.3. The subset L0 ⊂ Lp of the functions such that f(x) = 0 for µ–almost all x ∈ M is easily seen to be a subspace; the corresponding factor space Lp (M, dµ) := Lp (M, dµ)/L0 is then formed by the classes of µ–equivalent functions.
- 26. 1.1 Vector and normed spaces 3 A map f : V → C on a vector space V is called a functional; if it maps into the reals we speak about a real functional. A functional f is additive if f(x + y) = f(x)+ f(y) holds for all x, y ∈ V , and homogeneous if f(αx) = αf(x) or antihomogeneous if f(αx) = ᾱf(x) for x ∈ V, α ∈ C. An additive (anti)homogeneous functional is called (anti)linear. A real functional p is called a seminorm if p(x + y) ≤ p(x)+p(y) and p(αx) = |α|p(x) holds for any x, y ∈ V, α ∈ C; this definition implies that p maps V into R+ and |p(x)−p(y)| ≤ p(x−y). The following important result is valid (see the notes to this chapter). 1.1.3 Theorem (Hahn–Banach): Let p be a seminorm on a vector space V . Any linear functional f0 defined on a subspace V0 ⊂ V and fulfilling |f0(y)| ≤ p(y) for all y ∈ V0 can be extended to a linear functional f on V such that |f(x)| ≤ p(x) holds for any x ∈ V . A map F := V × · · · × V → C is called a form, in particular, a real form if its range is contained in R. A form F : V × V → C is bilinear if it is linear in both arguments, and sesquilinear if it is linear in one of them and antilinear in the other. Most frequently we shall drop the adjective when speaking about sesquilinear forms; we shall use the “physical” convention assuming that such a form is antilinear in the left argument. For a given F we define the quadratic form (generated by F ) by qF : qF (x) = F(x, x); the correspondence is one–to–one as the polarization formula F(x, y) = 1 4 qF (x+y) − qF (x−y) − i 4 qF (x+iy) − qF (x−iy) shows. A form F is symmetric if F(x, y) = F(y, x) for all x, y ∈ V ; it is positive if qF (x) ≥ 0 for any x ∈ V and strictly positive if, in addition, F(x) = 0 holds for x = 0 only. A positive form is symmetric (Problem 6) and fulfils the Schwarz inequality, |F(x, y)|2 ≤ qF (x)qF (y) . A norm on a vector space V is a seminorm · such that x = 0 holds iff x = 0. A pair (V, · ) is called a normed space; if there is no danger of misunderstanding we shall speak simply about a normed space V . 1.1.4 Examples: (a) In the spaces Cn and Rn , we introduce x ∞ := max 1≤j≤n |ξj| and x p := n j=1 |ξj|p 1/p , p ≥ 1 , for x = {ξ1, . . . , ξn}; the norm · 2 on Rn is often also denoted as | · |. Analogous norms are used in p (see also Problem 8). (b) In Lp (M, dµ), we introduce f p := M |f|p dµ 1/p .
- 27. 4 1 Some notions from functional analysis The relation f p =0 implies f(x)=0 µ–a.e. in M, so f is the zero element of Lp (M, dµ). If we speak about Lp (M, dµ) as a normed space, we always have in mind this natural norm though it is not, of course, the only possibility. If the measure µ is discrete with countable support, Lp (M, dµ) is isomorphic to p and we recover the norm · p of the previous example. (c) By L∞ (M, dµ) we denote the set of classes of µ–equivalent functions f : M → C, which are bounded a.e., i.e., there is c 0 such that |f(x)| ≤ c for µ–almost all x ∈ M. The infimum of all such numbers is denoted as sup ess |f(x)|. We can easily check that L∞ (M, dµ) is a vector space and f → f ∞ := sup ess x∈M |f(x)| is a norm on it. (d) The space C(X) can be equipped with the norm f ∞ := supx∈X |f(x)|. A strictly positive sesquilinear form on a vector space V is called an inner (or scalar) product. In other words, it is a map (·, ·) from V × V to C such that the following conditions hold for any x, y, z ∈ V and α ∈ C: (i) (x, αy+z) = α(x, y) + (x, z) (ii) (x, y) = (y, x) (iii) (x, x) ≥ 0 and (x, x)=0 iff x=0 A vector space with an inner product is called a pre–Hilbert space. Any such space is at the same time a normed space with the norm x := (x, x); the Schwarz inequality then assumes the form |(x, y)| ≤ x y . The above norm is said to be induced by the inner product. Due to conditions (i) and (ii) it fulfils the parallelogram identity, x+y 2 + x−y 2 = 2 x 2 + 2 y 2 ; on the other hand, it allows us to express the inner product by polarization, (x, y) = 1 4 x+y 2 − x−y 2 − i 4 x+iy 2 − x−iy 2 . These properties are typical for a norm induced by an inner product (Problem 11). Vectors x, y of a pre–Hilbert space V are called orthogonal if (x, y) = 0. A vector x is orthogonal to a set M if (x, y) = 0 holds for all y ∈ M; the set of all such vectors is denoted as M⊥ and called the orthogonal complement to M. Inner– product linearity implies that it is a subspace, (M⊥ )lin = M⊥ , with the following simple properties (Mlin)⊥ = M⊥ , Mlin ⊂ (M⊥ )⊥ , M ⊂ N ⇒ M⊥ ⊃ N⊥ .
- 28. 1.2 Metric and topological spaces 5 A set M of nonzero vectors whose every two elements are orthogonal is called an orthogonal set; in particular, M is orthonormal if x = 1 for each x ∈ M. Any orthonormal set is obviously linearly independent, and in the opposite direction we have the following assertion, the proof of which is left to the reader. 1.1.5 Theorem (Gram-Schmidt): Let N be an at most countable linearly inde- pendent set in a pre–Hilbert space V , then there is an orthonormal set M ⊂ V of the same cardinality such that Mlin = Nlin. 1.2 Metric and topological spaces A metric on a set X is a map : X × X → [0, ∞), which is symmetric, (x, y) = (y, x), (x, y) = 0 iff x = y, and fulfils the triangle inequality, (x, z) ≤ (x, y) + (y, z) , for any x, y, z ∈ X; the pair (X, ) is called a metric space (we shall again for simplicity often use the symbol X only). If X is a normed space, one can define a metric on X by (x, y) := x−y ; we say it is induced by the norm (see also Problems 15 and 16). Let us first recall some basic notions and properties of metric spaces. An ε– neighborhood of a point x ∈ X is the open ball Uε(x) := { y ∈ X : (y, x) ε}. A point x is an interior point of a set M if there is a Uε(x) ⊂ M. A set is open if all its points are interior points, in particular, any neighborhood of a given point is open. A union of an arbitrary family of open sets is again an open set; the same is true for finite intersections of open sets. The closure M of a set M is the family of all points x ∈ X such that the intersection Uε(x) ∩ M = ∅ for any ε 0. A point x ∈ M is called isolated if there is Uε(x) such that Uε(x)∩M = {x}, otherwise x is a limit (or accumulation) point of M. The closure points of M which are not interior form the boundary bd M of M. A set is closed if it coincides with its closure, and M is the smallest closed set containing M (cf. Problem 17). In particular, the whole X and the empty set ∅ are closed and open at the same time. A set M is said to be dense in a set N ⊂ X if M ⊃ N ; it is everywhere dense if M = X and nowhere dense if X M is everywhere dense. A metric space which contains a countable everywhere dense set is called separable. An example is the space Cn with any of the norms of Example 1.1.4a where a dense set is formed, e.g., by n–tuples of complex numbers with rational real and imaginary parts; other examples will be given in the next chapter (see also Problem 18). A sequence {xn} ⊂ X converges to a point x ∈ X if to any Uε(x) there is n0 such that xn ∈ Uε(x) holds for all n n0. Since any two mutually different points x, y ∈ X have disjoint neighborhoods, each sequence has at most one limit. Sequences can also be used to characterize closure of a set (Problem 17). Next we recall a few notions related to maps f : X → X of metric spaces. The map f is continuous at a point x ∈ X if to any U ε(f(x)) there is a Uδ(x) such
- 29. 6 1 Some notions from functional analysis that f(Uδ(x)) ⊂ U ε(f(x)); alternatively we can characterize the local continuity using sequences (Problem 19). On the other hand, f is (globally) continuous if the pull–back f(−1) (G ) of any open set G ⊂ X is open in X. An important class of continuous maps is represented by homeomorphisms, i.e., bijections f : X → X such that both f and f−1 are continuous. It is clear that in this way any family of metric spaces can be divided into equivalence classes. A homeomorphism maps, in particular, the family τ of open sets in X bijectively onto the family τ of open sets in X ; we say that homeomorphic metric spaces are topologically equivalent. Such spaces can still differ in metric properties. As an example, consider the spaces R and (−π 2 , π 2 ) with the same metric (x, y) := |x−y|; they are homeomorphic by x → arctan x but only the first of them contains un- bounded sets. A bijection f := X → X which preserves the metric properties, (f(x), f(y)) = (x, y), is called isometry; this last named property implies conti- nuity, so any isometry is a homeomorphism. A homeomorphism f : V → V of normed spaces is called linear homeomor- phism if it is simultaneously an isomorphism. Linearly homeomorphic spaces there- fore also have the same algebraic structure; this is particularly simplifying in the case of finite dimension (Problem 21). In addition, if the identity f(x) V = x V holds for any x ∈ V we speak about a linear isometry. A sequence {xn} in a metric space X is called Cauchy if to any ε 0 there is nε such that (xn, xm) ε for all n, m nε. In particular, any convergent sequence is Cauchy; a metric space in which the converse is also true is called complete. Completeness is one of the basic “nontopological” properties of metric spaces: recall the spaces R and (−π 2 , π 2 ) mentioned above; they are homeomorphic but only the first of them is complete. 1.2.1 Example: Let us check the completeness of Lp (M, dµ) , p ≥ 1, with a σ–finite measure µ. Suppose first µ(M) ∞ and consider a Cauchy sequence {fn} ⊂ Lp . By the Hölder inequality, it is Cauchy also with respect to · 1, so for any ε 0 there is N(ε) such that fn −fm 1 ε for n, m N(ε). We pick a subsequence, gn := fkn , by choosing k1 := N(2−1 ) and kn+1 := max{kn +1, N(2−n−1) )}, so gn+1−gn 1 2−n , and the functions ϕn := |g1| + n−1 =1 |g+1−g| obey M ϕndµ ≤ g1 1 + n−1 =1 2− 1 + g1 1 . Since they are measurable and form a nondecreasing sequence, the monotone– convergence theorem implies existence of a finite limn→∞ ϕn(x) for µ–a.a. x ∈ M. Furthermore, |gn+p−gn| ≤ ϕn+p −ϕn, so there is a function f which is finite µ-a.e. in M and fulfils f(x) = limn→∞ gn(x). The sequence {gn} has been picked from a Cauchy sequence and it is therefore Cauchy also, gn−gm p ε for all n, m Ñ(ε) for a suitable Ñ(ε). On the other hand, limm→∞ |gn(x)−gm(x)|p = |gn(x)−f(x)|p for µ–a.a. x ∈ M, so Fatou’s lemma implies gn−f p ≤ ε for all n Ñ(ε); hence f ∈ Lp and limn→∞ fn−f p = 0 (Problem 24).
- 30. 1.2 Metric and topological spaces 7 If µ is σ–finite and µ(M) = ∞, there is a disjoint decomposition ∞ j=1 Mj = M with µ(Mj) ∞. The already proven completeness of Lp (Mj, dµ) implies the existence of functions f(j) ∈ Lp (Mj, dµ) which fulfil f (j) n −f(j) p → 0 as n → ∞; then we can proceed as in the proof of completeness of p (cf. Problem 23). Other examples of complete metric spaces are given in Problem 23. Any metric space can be extended to become complete: a complete space (X , ) is called the completion of (X, ) if (i) X ⊂ X and (x, y) = (x, y) for all x, y ∈ X, and (ii) the set X is everywhere dense in X (this requirement ensures minimality — cf. Problem 25). 1.2.2 Theorem: Any metric space (X, ) has a completion. If (X̃, ˜ ) is another completion of (X, ), there is an isometry f : X → X̃ which preserves X, i.e., f(x) = x for all x ∈ X. Sketch of the proof: Uniqueness follows directly from the definition. Existence is proved constructively by the so–called standard completion procedure which genera- lizes the Cantor construction of the reals. We start from the set of all Cauchy sequences in (X, ). This can be factorized if we set {xj} ∼ {yj} for the sequences with limj→∞ (xj, yj) = 0. The set of equivalence classes we denote as X∗ and define ∗ ([x], [y]) := limj→∞ (xj, yj) to any [x], [y] ∈ X∗ . Finally, one has to check that this definition makes sense, i.e., that ∗ does not depend on the choice of sequences representing the classes [x], [y], ∗ is a metric on X∗ , and (X∗ , ∗ ) satisfies the requirements of the definition. The notion of topology is obtained by axiomatization of some properties of metric spaces. Let X be a set and τ a family of its subsets which fulfils the following conditions (topology axioms): (t1) X ∈ τ and ∅ ∈ τ. (t2) If I is any index set and Gα ∈ τ for all α ∈ I; then α∈I Gα ∈ τ. (t3) n j=1 Gj ∈ τ for any finite subsystem {G1, . . . , Gn} ⊂ τ. The family τ is called a topology, its elements open sets and the set X equipped with a topology is a topological space; when it is suitable we write (X, τ). A family of open sets in a metric space (X, ) is a topology by definition; we speak about the metric–induced topology τ, in particular, the norm–induced topology if X is a vector space and is induced by a norm. On the other hand, finding the conditions under which a given topology is induced by a metric is a nontrivial problem (see the notes). Two extreme topologies can be defined on any set X: the discrete topology τd := 2X , i.e., the family of all subsets in X, and the trivial topology τ0 := {∅, X}. The first of them is induced by the discrete metric, d(x, y) := 1 for x=y, while (X, τ0) is not metrizable unless X is a one–point set. An open set in a topological space X containing a point x or a set M ⊂ X is called a neighborhood of the point X or the set M, respectively. Using this concept,
- 31. 8 1 Some notions from functional analysis we can adapt to topological spaces most of the “metric” definitions presented above, as well as some simple results such as those of Problems 17a, c, 19b, topological equivalence of homeomorphic spaces, etc. On the other hand, equally elementary metric–space properties may not be valid in a general topological space. 1.2.3 Example: Consider the topologies τfin and τcount on X = [0, 1] in which the closed sets are all finite and almost countable subsets of X, respectively. If {xn} ⊂ X is a simple sequence, xn = xm for n = m ; then any neighborhood U(x) contains all elements of the sequence with the exception of a finite number; hence the limit is not unique in (X, τfin). This is not the case in (X, τcount) but there only very few sequences converge, namely those with xn = xn0 for all n ≥ n0, which means, in particular, that we cannot use sequences to characterize local continuity or points of the closure. Some of these difficulties can be solved by introducing a more general notion of convergence. A partially ordered set I is called directed if for any α, β ∈ I there is γ ∈ I such that α ≺ γ and β ≺ γ. A map of a directed index set I into a topological space X, α → xα, is called a net in X. A net {xα} is said to converge to a point x ∈ X if to any neighborhood U(x) there is an α0 ∈ I such that xα ∈ U(x) for all α α0. To illustrate that nets in a sense play the role that sequences played in metric spaces, let us mention two simple results the proofs of which we leave to the reader (Problem 29). 1.2.4 Proposition: Let (X, τ) , (X , τ ) be topological spaces; then (a) A point x ∈ X belongs to the closure of a set M ⊂ X iff there is a net {xα} ⊂ M such that xα → x. (b) A map f : X → X is continuous at a point x ∈ X iff the net {f(xα)} converges to f(x) for any net {xα} converging to x. Two topologies can be compared if there is an inclusion between them, τ1 ⊂ τ2, in which case we say that τ1 is weaker (coarser) than τ2; while the latter is stronger (finer) than τ1. Such a relation between topologies has some simple consequences — see, e.g., Problem 32. In particular, continuity of a map f : X → Y is preserved when we make the topology in Y weaker or in X stronger. In other cases it may not be preserved; for instance, Problem 3.9 gives an example of three topologies, τw ⊂ τs ⊂ τu, on a set X := B(H) and a map f : X → X which is continuous with respect to τw and τu but not τs. 1.2.5 Example: A frequently used way to construct a topology on a given X employs a family F of maps from X to a topological space (X̃, τ̃). Among all topologies such that each f ∈ F is continuous there is one which is the weakest; its existence follows from Problem 30, where the system S consists of the sets f(−1) G) for each G ⊂ τ̃, f ∈ F. We call this the F–weak topology.
- 32. 1.2 Metric and topological spaces 9 For any set M in a topological space (X, τ) we define the relative topology τM as the family of intersections M ∩ G with G ⊂ τ; the space (M, τM ) is called a subspace of (X, τ). Other important notions are obtained by axiomatization of properties of open balls in metric spaces. A family B ⊂ τ is called a basis of a topological space (X, τ) if any nonempty open set can be expressed as a union of elements of B. A family Bx of neighborhoods of a given point x ∈ X is called a local basis at x if any neighborhood U(x) contains some B ∈ Bx. A trivial example of both a basis and a local basis is the topology itself; however, we are naturally more interested in cases where bases are rather a “small part” of it. It is easy to see that local bases can be used to compare topologies. 1.2.6 Proposition: Let a set X be equipped with topologies τ, τ with local bases Bx, B x at each x ∈ X. The inclusion τ ⊂ τ holds iff for any B ∈ Bx there is B ∈ B x such that B ⊂ B. To be a basis of a topology or a local basis, a family of sets must meet certain consistency requirements (cf. Problem 30c, d); this is often useful when we define a particular topology by specifying its basis. 1.2.7 Example: Let (Xj, τj), j = 1, 2, be topological spaces. On the Cartesian product X1 × X2 we define the standard topology τX1×X2 determined by τj, j = 1, 2, as the weakest topology which contains all sets G1 × G2 with Gj ∈ τj, i.e., τX1×X2 := τ(τ1 × τ2) in the notation of Problem 30b. Since (A1 × A2) ∩ (B1 × B2) = (A1 ∩ B1) × (A2 ∩ B2), the family τ1 × τ2 itself is a basis of τX1×X2 ; a local basis at [x1, x2] consists of the sets U(x1) × V (x2), where U(x1) ∈ τ1, V (x2) ∈ τ2 are neighborhoods of the points x1, x2, respectively. The space (X1 × X2, τX1×X2 ) is called the topological product of the spaces (Xj, τj), j = 1, 2. Bases can also be used to classify topological spaces by the so–called countability axioms. A space (X, τ) is called first countable if it has a countable local basis at any point; it is second countable if the whole topology τ has a countable basis. The second requirement is actually stronger; for instance, a nonseparable metric space is first but not second countable (cf. Problem 18; some related results are collected in Problem 31). The most important consequence of the existence of a countable local basis, {Un(x) : n = 1, 2, . . .} ⊂ τ, is that one can pass to another local basis {Vn(x) : n = 1, 2, . . .}, which is ordered by inclusion, Vn+1 ⊂ Vn, setting V1 := U1 and Vn+1 := Vn ∩ Un+1. This helps to partially rehabilitate sequences as a tool in checking topological properties (Problem 33a). The other problem mentioned in Example 1.2.3, namely the possible nonunique- ness of a sequence limit, is not related to the cardinality of the basis but rather to the degree to which a given topology separates points. It provides another classification of topological spaces through separability axioms: T1 To any x, y ∈ X, x=y, there is a neighborhood U(x) such that y ∈ U(x). T2 To any x, y ∈ X, x=y, there are disjoint neighborhoods U(x) and U(y).
- 33. 10 1 Some notions from functional analysis T3 To any closed set F ⊂ X and a point x ∈ F, there are disjoint neighborhoods U(x) and U(F). T4 To any pair of disjoint closed sets F, F , there are disjoint neighborhoods U(F) and U(F ). A space (X, τ) which fulfils the axioms T1 and Tj is called Tj–space, T2–spaces are also called Hausdorff, T3–spaces are regular, and T4–spaces are normal . For instance, the spaces of Example 3 are T1 but not Hausdorff; one can find examples showing that the whole hierarchy is nontrivial (see the notes). In particular, any metric space is normal. The question of limit uniqueness that we started with is answered affirmatively in Hausdorff spaces (see Problem 29). 1.3 Compactness One of the central points in an introductory course of analysis is the Heine–Borel the- orem, which claims that given a family of open intervals covering a closed bounded set F ⊂ R, we can select a finite subsystem which also covers F. The notion of compactness comes from axiomatization of this result. Let M be a set in a topo- logical space (X, τ). A family P := {Mα : α ∈ I} ⊂ 2X is a covering of M if α∈I Mα ⊃ M; in dependence on the cardinality of the index set I the covering is called finite, countable, etc.We speak about an open covering if P ⊂ τ. The set M is compact if an arbitrary open covering of M has a finite subsystem that still covers M; if this is true for the whole of X we say that the topological space (X, τ) is compact. It is easy to see that compactness of M is equivalent to com- pactness of the space (M, τM ) with the induced topology, so it is often sufficient to formulate theorems for compact spaces only. 1.3.1 Proposition: Let (X, τ) be a compact space, then (a) Any infinite set M ⊂ X has at least one accumulation point. (b) Any closed set F ⊂ X is compact. (c) If a map f : (X, τ) → (X , τ ) is continuous, then f(X) is compact in (X , τ ). Proof: To check (a) it is obviously sufficient to consider countable sets. Suppose M = {xn : n = 1, 2, . . .} has no accumulation points; then the same is true for the sets MN := {xn : n ≥ N}. They are therefore closed and their complements form an open covering of X with no finite subcovering. Further, let {Gα} be an open covering of F; adding the set G := X F we get an open covering of X. Any finite subcovering G of X is either contained in {Gα} or it contains the set G; in the latter case G G is a finite covering of the set F. Finally, the last assertion follows from the appropriate definitions.
- 34. 1.3 Compactness 11 Part (a) of the proposition represents a particular case of a more general result (see the notes) which can be used to define compactness; another alternative defini- tion is given in Problem 36. Compactness has an important implication for the way in which the topology separates points. 1.3.2 Theorem: A compact Hausdorff space is normal. Proof: Let F, R be disjoint closed sets and y ∈ R. By assumption, to any x ∈ F one can find disjoint neighborhoods Uy(x) and Ux(y). The family {Uy(x) : x ∈ F} covers the set F, which is compact in view of the previous proposition; hence there is a finite subsystem {Uy(xj) : j = 1, . . . , n} such that Uy(F) := n j=1 Uy(xj) is a neighborhood of F. Moreover, U(y) := n j=1 Uxj (y) is a neighborhood of the point y and U(y) ∩ Uy(F) = ∅. This can be done for any point y ∈ R giving an open covering {U(y) : y ∈ R} of the set R; from it we select again a finite subsystem {U(yk) : k = 1, . . . , m} such that U(R) := m k=1 U(yk) is a neighborhood of R which has an empty intersection with U(F) := m k=1 Uyk (F). 1.3.3 Theorem: Let X be a Hausdorff space, then (a) Any compact set F ⊂ X is closed. (b) If the space X is compact, then any continuous bijection f : X → X for X Hausdorff is a homeomorphism. Proof: If y ∈ F, the neighborhood U(y) from the preceding proof has an empty intersection with F, so y ∈ F. To prove (b) we have to check that f(F) is closed in X for any closed F ⊂ X; this follows easily from (a) and Proposition 1.3.1c. A set M in a topological space is called precompact (or relatively compact) if M is compact. A space X is locally compact if any point x ∈ X has a precompact neighborhood; it is σ–compact if any countable covering has a finite subcovering. Let us now turn to compactness in metric spaces. There, any compact set is closed by Theorem 1.3.3 and bounded — from an unbounded set we can always select an infinite subset which has no accumulation point. However, these conditions are not sufficient. For instance, the closed ball S1(0) in 2 is bounded but not compact: its subset consisting of the points Xj := {δjk}∞ k=1 , j = 1, 2, . . ., has no accumulation point because Xj −Xk = √ 2 holds for all j = k. To be able to characterize compactness by metric properties we need a stronger condition. Given a set M in a metric space (X, ) and ε 0, we call a set Nε an ε–lattice for M if to any x ∈ M there is a y ∈ Nε such that (x, y) ≤ ε ( Nε may not be a subset of M but by using it one is able to construct a 2ε– lattice for M which is contained in M ). A set M is completely bounded if it has a finite ε–lattice for any ε 0; if the set X itself is completely bounded we speak about a completely bounded metric space. If M is completely bounded, the same is obviously true for M. Any completely bounded set is bounded; on the other hand, any infinite orthonormal set in a pre–Hilbert space represents an example of a set which is bounded but not completely bounded.
- 35. 12 1 Some notions from functional analysis 1.3.4 Proposition: A σ–compact metric space is completely bounded. A completely bounded metric space is separable. Proof: Suppose that for some ε 0 there is no finite ε–lattice. Then X Sε(x1) = ∅ for an arbitrarily chosen x1 ∈ X, otherwise {x1} would be an ε–lattice for X. Hence there is x2 ∈ X such that (x1, x2) ε and we have X (Sε(x1)∪Sε(x2)) = ∅ etc.; in this way we construct an infinite set {xj : j = 1, 2, . . .} which fulfils (xj, xk) ε for all j = k, and therefore it has no accumulation points. As for the second part, if Nn is a (1/n)–lattice for X, then ∞ n=1 Nn is a countable everywhere dense set. 1.3.5 Corollary: Let X be a metric space; then the following conditions are equiv- alent: (i) X is compact. (ii) X is σ–compact. (iii) Any infinite set in X has an accumulation point. 1.3.6 Theorem: A metric space is compact iff it is complete and completely bounded. Proof: Let X be compact; in view of Proposition 1.3.4 it is sufficient to show that it is complete. If {xn} is Cauchy, the compactness implies existence of a convergent subsequence so {xn} is also convergent (Problem 24). On the other hand, to prove the opposite implication we have to check that any M := {xn : n = 1, 2, . . .} ⊂ X has an accumulation point. By assumption, there is a finite 1–lattice N1 for X, hence there is y1 ∈ N1 such that the closed ball S1(y1) contains an infinite subset of M. The ball S1(y1) is completely bounded, so we can find a finite (1/2)–lattice N2 ⊂ S1(y1) and a point y2 ∈ N2 such that the set S1/2(y2) ∩ M is infinite. In this way we get a sequence of closed balls Sn := S21 − n (yn) such that each of them contains infinitely many points of M and their centers fulfil yn+1 ∈ Sn. The closed balls of doubled radii then satisfy S21 − n (yn+1) ⊂ S22 − n (yn) and M has an accumulation point in view of Problem 26. 1.3.7 Corollary: (a) A set M in a complete metric space X is precompact iff it is completely bounded. In particular, if X is a finite–dimensional normed space, then M is precompact iff it is bounded. (b) A continuous real–valued function f on a compact topological space X is bounded and assumes its maximum and minimum values in X. Proof: The first assertion follows from Problem 25. If M is compact, it is bounded so M is also bounded. To prove the opposite implication in a finite–dimensional normed space, we can use the fact that such a space is topologically isomorphic to Cn (or Rn in the case of a real normed space — see Problem 21). As for part (b), the set f(X) ⊂ R is compact by Proposition 1.3.1c, and therefore bounded. Denote
- 36. 1.4 Topological vector spaces 13 α := supx∈X f(x) and let {xn} ⊂ X be a sequence such that f(xn) → α. Since X is compact there is a subsequence {xkn } converging to some xs and the continuity implies f(xs) = α. In the same way we can check that f assumes a minimum value. 1.4 Topological vector spaces We can easily check that the operations of summation and scalar multiplication in a normed space are continuous. Let us now see what would follow from such a requirement when we combine algebraic and topological properties. A vector space V equipped with a topology τ is called a topological vector space if (tv1) The summation maps continuously (V × V, τV ×V ) to (V, τ). (tv2) The scalar multiplication maps continuously (C×V, τC×V ) to (V, τ). (tv3) (V, τ) is Hausdorff. In the same way, we define a topological vector space over any field. Instead of (tv3), we may demand T1–separability only because the first two requirements imply that T3 is valid (Problem 39). A useful tool in topological vector spaces is the family of translations, tx : V → V , defined for any x ∈ V by tx(y) := x + y. Since t−1 x = t−x, the continuity of summation implies that any translation is a homeomorphism; hence if G is an open set, then x + G := tx(G) is open for all x ∈ V ; in particular, U is a neighborhood of a point x iff U = x + U(0), where U(0) is a neighborhood of zero. This allows us to define a topology through its local basis at a single point (Problem 40). Suppose a map between topological vector spaces (V, τ) and (V , τ ) is simul- taneously an algebraic isomorphism of V, V and a homeomorphism of the corre- sponding topological spaces, then we call it a linear homeomorphism (or topological isomorphism). As in the case of normed spaces (cf. Problem 21), the structure of a finite–dimensional topological vector space is fully specified by its dimension. 1.4.1 Theorem: Twofinite–dimensionaltopologicalvectorspaces, (V, τ) and (V , τ ), are linearly homeomorphic iff dim V = dim V . Any finite–dimensional topological vector space is locally compact. Proof: It is sufficient to construct a linear homeomorphism of a given n–dimensional (V, τ) to Cn . We take a basis {e1, . . . , en} ⊂ V and construct f : V → Cn by f n j=1 ξjej := [ξ1, . . . , ξn]; in view of the continuity of translations we have to show that f and f−1 are continuous at zero. According to (tv1), for any U(0) ∈ τ we can find neighborhoods Uj(0) such that n j=1 xj ∈ U(0) for xj ∈ Uj(0) , j = 1, . . . , n and f−1 is continuous by Problem 42a. To prove that f is continuous we use the fact that V is Hausdorff: Proposition 1.3.1 and Theorem 1.3.3 together with the already proven continuity of f−1 ensure that Sε := {x∈V : f(x) = ε} = f(−1) (Kε) is closed for any ε 0; we have denoted here by Kε the ε–sphere
- 37. 14 1 Some notions from functional analysis in Cn . Since 0 ∈ Sε, the set G := V Sε is a neighborhood of zero, and by Problem 42b there is a balanced neighborhood U ⊂ G of zero; this is possible only if f(x) ε for all x ∈ U. Next we want to discuss a class of topological vector spaces whose properties are closer to those of normed spaces. In distinction to the latter the topology in them is not specified generally by a single (semi)norm but rather by a family of them. Let P := {pα : α ∈ I} be a family of seminorms on a vector space V where I is an arbitrary index set. We say that P separates points if to any nonzero x ∈ V there is a pα ∈ P such that pα(x) = 0. It is clear that if P consists of a single seminorm p it separates points iff p is a norm. Given a family P we set Bε(p1, . . . , pn) := { x ∈ V : pj(x) ε , j = 1, . . . , n } ; the collection of these sets for any ε 0 and all finite subsystems of P will be denoted as BP 0 . In view of Problem 40, BP 0 defines a topology on V which we denote as τP . 1.4.2 Theorem: If a family P of seminorms on a vector space V separates points, then (V, τP ) is a topological vector space. Proof: By assumption, to a pair x, y of different points there is a p ∈ P such that ε := 1 2 p(x − y) 0. Then U(x) := x + Bε(p) and U(y) := y + Bε(p) are disjoint neighborhoods, so the axiom T2 is valid. The continuity of summation at the point [0, 0] follows from the inequality p(x+y) ≤ p(x)+p(y); for the scalar multiplication we use p(αx−α0x0) ≤ |α − α0| p(x0) + |α| p(x−x0). A topological vector space with a topology induced by a family P separat- ing points is called locally convex. This name has an obvious motivation: if x, y ∈ Bε(p1, . . . , pn), then pj(tx+(1−t)x) ≤ tpj(x)+(1−t)pj(x) holds for any t ∈ [0, 1] so the sets Bε(p1, . . . , pn) are convex. The convexity is preserved at translations, so the local basis of τP at each x ∈ V consists of convex sets (see also the notes). 1.4.3 Example: The family P := {px := |(x, ·)| : x ∈ V } in a pre–Hilbert space V generates a locally convex topology which is called the weak topology and is denoted as τw; it is easy to see that it is weaker than the “natural” topology induced by the norm. 1.4.4 Theorem: A locally convex space (V, τ) is metrizable iff there is a countable family P of seminorms which generates the topology τ. Proof: If V is metrizable it is first countable. Let {Uj : j = 1, 2, . . .} be a local basis of τ at the point 0. By definition, to any Uj we can find ε 0 and a finite subsystem Pj ⊂ P such that p∈Pj Bε(p) ⊂ Uj. The family P := ∞ j=1 Pj is countable and generates a topology τP which is not stronger than τ := τP ; the above inclusion shows that τP = τ. On the other hand, suppose that τ is generated by a family {pn : n = 1, 2, . . .} separating points; then we can define a metric as in Problem 16 and show that the corresponding topology satisfies τ = τ (Problem 43).
- 38. 1.5 Banach spaces and operators on them 15 A locally convex space which is complete with respect to the metric used in the proof is called a Fréchet space (see also the notes). 1.4.5 Example: The set S(Rn ) consists of all infinitely differentiable functions f : Rn → C such that f J,K := sup x∈Rn | xJ (DK f)(x)| ∞ holds for any multi–indices J := [j1, . . . , jn] , K := [k1, . . . , kn] with jr, kr non– negative integers, where xJ := ξj1 1 . . . ξjn n , DK := ∂|K| /∂ξk1 1 . . . ∂ξkn n and |K| := k1 + · · · + kn. It is easy to see that any such f and any derivative DK f (as well as polynomial combinations of them) tend to zero faster than |xJ |−1 for each J; we speak about rapidly decreasing functions. It is also clear that any · J,K is a seminorm, with f 0,0 = f ∞, and the family P := { · J,K} separates points. The corresponding locally convex space S(Rn ) is called the Schwartz space; one can show that it is complete, i.e., a Fréchet space (see the notes). An important subspace in S(Rn ) consists of infinitely differentiable functions with a compact support; we denote it as C∞ 0 (Rn ). It is dense, C∞ 0 (Rn) = S(Rn ) , (1.1) with respect to the topology of S(Rn ) (Problem 44). 1.5 Banach spaces and operators on them A normed space which is complete with respect to the norm–induced metrics is called a Banach space. We have already met some frequently used Banach spaces — see Example 1.2.1 and Problem 23. In view of Problem 21, any finite–dimensional normed space is complete; in the general case we have the following completeness criterion, the proof of which is left to the reader (see also Example 1.5.3b below). 1.5.1 Theorem: A normed space V is complete iff to any sequence {xn} ⊂ V such that ∞ n=1 xn ∞ there is an x ∈ V such that x = limn→∞ n k=1 xk (or in short, iff any absolutely summable sequence is summable). Given a noncomplete norm space, we can always extend it to a Banach space by the standard completion procedure (Problem 46). A set M in a Banach space X is called total if Mlin = X. Such a set is a basis if M is linearly independent and dim X ∞, while an infinite–dimensional space can contain linearly independent total sets, which are not Hamel bases of X (cf. the notes to Section 1.1). 1.5.2 Lemma: (a) If M is total in a Banach space X, then any set N ⊂ X dense in M is total in X. (b) A Banach space which contains a countable total set is separable.
- 39. 16 1 Some notions from functional analysis Proof: Part (a) follows from the appropriate definitions. Suppose that M = {x1, x2, . . .} is total in X and Crat is the countable set of complex numbers with ra- tional real and imaginary parts; then the set L := { n j=1 γjxj : γj ∈ Crat, n ∞} is countable. Since Crat is dense in C, we get L = X. 1.5.3 Examples: (a) The set P(a, b) of all complex polynomials on (a, b) is an infinite–dimensional subspace in C[a, b] := C([a, b]). By the Weierstrass theo- rem, any f ∈ C[a, b] can be approximated by a uniformly convergent sequence of polynomials; hence C[a, b] is a complete envelope of (P(a, b), · ∞). The set {xk : k = 0, 1, . . . } is total in C[a, b], which is therefore separable. (b) Consider the sequences Ek := {δjk}∞ j=1 in p , p ≥ 1. For a given X := {ξj} ∈ p , the sums Xn := n j=1 ξjEj are nothing else than truncated sequences, so limn→∞ X−Xn p = 0. Hence {Ek : k = 1, 2 . . .} is a countable total set and p is separable. Notice also that the sequence {ξjEj}∞ j=1 is summable but it may not be absolutely summable for p 1. (c) Consider next the space Lp (Rn , dµ) with an arbitrary Borel measure µ on Rn . We use the notation of Appendix A. In particular, J n is the family of all bounded intervals in Rn ; then we define S(n) := {χJ : J ∈ J n }. It is a subset in Lp and the elements of its linear envelope are called step functions; we can check that S(n) is total in Lp (Rn , dµ) (Problem 47). Combining this result with Lemma 1.5.2 we see that the subspace C∞ 0 (Rn ) is dense in Lp (Rn , dµ); in particular, for the Lebesgue measure on Rn the inclusions C∞ 0 (Rn ) ⊂ S(Rn ) ⊂ Lp (Rn ) yield (C∞ 0 (Rn))p = (S(Rn))p = Lp (Rn ) . (1.2) (d) Given a topological space (X, τ) we call C∞(X) the set of all continuous functions on X with the following property: for any ε 0 there is a compact set K ⊂ X such that |f(x)| ε outside K. It is not difficult to check that C∞(X) is a closed subspace in C(X) and C0(X) = C∞(X), where C0(X) is the set of continuous functions with compact support (Problem 48). In the particular case X = Rn , C∞ 0 (Rn ) is dense in C∞(Rn ) (see the notes), so (C∞ 0 (Rn))∞ = (S(Rn))∞ = C∞(Rn ) . (1.3) There are various ways in, which it is possible to construct new Banach spaces from given ones. We mention two of them (see also Problem 49): (i) Let {Xj : j = 1, 2, . . .} be a countable family of Banach spaces. We denote by X the set of all sequences x := {xj} , xj ∈ Xj, such that j xj j ∞ , and equip it with the “componentwise” defined summation and scalar mul- tiplication. The norm X ⊕ := j xj j turns it into a Banach space; the completeness can be checked as for p (Problem 23). The space (X, · ⊕) is called the direct sum of the spaces Xj , j = 1, 2, . . ., and denoted as ⊕ j Xj.
- 40. 1.5 Banach spaces and operators on them 17 (ii) Starting from the same family {Xj : j = 1, 2, . . .}, one can define another Banach space (which is sometimes also referred to as a direct sum) if we change the above norm to X ∞ := supj xj j replacing, of course, X by the set of sequences for which X ∞ ∞. The two Banach spaces are different unless the family {Xj} is finite; the present construction can easily be adapted to families of any cardinality. A map B : V1 → V2 between two normed spaces is called an operator; in particular, it is called a linear operator if it is linear. In this case we conventionally do not use parentheses and write the image of a vector x ∈ V1 as Bx. In this book we shall deal almost exclusively with linear operators, and therefore the adjective will usually be dropped. A linear operator B : V1 → V2 is said to be bounded if there is a positive c such that Bx 2 ≤ c x 1 for all x ∈ V1; the set of all such operators is denoted as B(V1, V2) or simply B(V ) if V1 = V2 := V . One of the elementary properties of linear operators is the equivalence between continuity and boundedness (Problem 50). The set B(V1, V2) becomes a vector space if we define on it summation and scalar multiplication by (αB + C)x := αBx + Cx. Furthermore, we can associate with every B ∈ B(V1, V2) the non–negative number B := sup S1 Bx 2 , where S1 := {x ∈ V1 : x 1 = 1 } is the unit sphere in V1 (see also Problem 51). 1.5.4 Proposition: The map B → B is a norm on B(V1, V2). If V2 is complete, the same is true for B(V1, V2), i.e., it is a Banach space. Proof: The first assertion is elementary. Let {Bn} be a Cauchy sequence in B(V1, V2); then for all n, m large enough we have Bn−Bm ε, and therefore Bnx−Bmx 2 ≤ ε x 1 for any x ∈ V1. As a Cauchy sequence in V2, {Bnx} converges to some B(x) ∈ V2. The linearity of the operators Bn implies that x → Bx is linear, B(x) = Bx. The limit m → ∞ in the last inequality gives Bx−Bnx 2 ≤ ε x 1, so B ∈ B(V1, V2) by the triangle inequality, and B−Bn ≤ ε for all n large enough. The norm on B(V1, V2) introduced above is called the operator norm. It has an additional property: if C : V1 → V2 and B : V2 → V3 are bounded operators, and BC is the operator product understood as the composite mapping V1 → V3, we have B(Cx) 3 ≤ B Cx 2 ≤ B C x 1 for all x ∈ V1, so BC is also bounded and BC ≤ B C . (1.4) Let V1 be a subspace of a normed space Ṽ1. An operator B : V1 → V2 is called a restriction of B̃ : Ṽ1 → V2 to the subspace V1 if Bx = B̃x holds for all x ∈ V1, and on the other hand, B̃ is said to be an extension of B; we write B = B̃ | V1 or B ⊂ B̃. Another simple property of bounded operators is that they can be extended uniquely by continuity.
- 41. 18 1 Some notions from functional analysis 1.5.5 Theorem: Assume that X1, X2 are Banach spaces and V1 is a dense subspace in X1; then any B ∈ B(V1, X2) has just one extension B̃ ∈ B(X1, X2), and moreover, B̃ = B . Proof: For any x ∈ X we can find a sequence {xn} ⊂ V1 that converges to x. Since B is bounded, the sequence {Bxn} is Cauchy, so there is a y ∈ X2 such that Bxn−y 2 → 0. We can readily check that y does not depend on the choice of the approximating sequence and the map x → y is linear; we denote it as B̃. If x ∈ V1 one can choose xn = x for all n, which means B̃ | V1 = B. Passing to the limit in the relation Bxn 2 ≤ B xn 1 we get B̃ ∈ B(X1, X2) and B̃ ≤ B ; since B̃ is an extension of B the two norms must be equal. Suppose finally that B = C | V1 for some C ∈ B(X1, X2). We have Cxn = Bxn , n = 1, 2, . . ., for any approximating sequence, and therefore C = B̃. Notice that in view of Proposition 1.5.4, B(V1, X) is complete even if V1 is not. The approximation procedure used in the proof to define B̃ is often of practical importance, namely if we study an operator whose action on some dense subspace is given by a simple formula. 1.5.6 Example (Fourier transformation): Following the usual convention we denote the scalar product of the vectors x, y ∈ Rn by x·y and set ˆ f(y) := (2π)−n/2 Rn e−i x·y f(x) dx and ˇ f(y) := ˆ f(−y) (1.5) for any f ∈ S(Rn ) and y ∈ Rn . The function ˆ f is well–defined and one can check that it belongs to S(Rn ) (Problem 52), i.e., that F0 : F0f = ˆ f is a linear map of S(Rn ) onto itself. We want to prove that F̃0 : f → ˇ f is its inverse. To this end, we use the relation from Problem 52; choosing gε := e−ε2|x|2/2 we get Rn ei x·y−ε2|x|2/2 ˆ f(y) dy = Rn e−|z|2/2 f(x+εz) dz for any ε 0. The two integrated functions can be majorized independently of ε; the limit ε → 0+ then yields F̃0F0f = f. Since ˇ f(x) = ˆ f(−x) we also get F0F̃0f = f for all f ∈ S(Rn ), i.e., the relation (F−1 0 f)(x) = (F0f)(−x) . Using Theorem 1.5.5, we shall now construct two important extensions of the oper- ator F0. For the moment, we denote by Sp(Rn ) the normed space (S(Rn ), · p); we know from (1.2) that it is dense in Lp (Rn ) , p ≥ 1. (i) Since S(Rn ) is a subset of C∞(Rn ) and ˆ f ∞ ≤ (2π)−n/2 f 1, the operator F0 can also be understood as an element of B(S1(Rn ), C∞(Rn )). As such it extends uniquely to the operator F ∈ B(L1 (Rn ), C∞(Rn )); it is easy to check that its action on any f ∈ L1 (Rn ) can be expressed again by the first one of the relations (1.5), (Ff)(y) = (2π)−n/2 Rn e−i x·y f(x) dx .
- 42. 1.5 Banach spaces and operators on them 19 The function Ff is called the Fourier transform of f. We have Ff ∈ C∞(Rn ), and therefore lim |x|→∞ (Ff)(x) = 0; this relation is often referred to as the Riemann-Lebesgue lemma. (ii) Using once more the relation from Problem 52, now with g := ˆ f = ˇ f, we find Rn | ˆ f(y)|2 dy = Rn |f(y)|2 dy for any f ∈ S(Rn ). This suggests another possible interpretation of the op- erator F0 as an element of B(S2(Rn ), L2 (Rn )). Extending it by continuity, we get the operator F ∈ B(L2 (Rn )), which is called the Fourier–Plancherel operator or briefly FP–operator; if it is suitable to specify the dimension of Rn we denote it as Fn. The above relation shows that Ffk 2 = fk 2 holds for the elements of any sequence {fk} ⊂ S2 approximating a given f ∈ L2 , and therefore Ff 2 = f 2 for all f ∈ L2 (Rn ); this implies that F is surjective (Problem 53). Hence the Fourier–Plancherel operator is a linear isometry of L2 (Rn ) onto itself. We are naturally interested in how F acts on the vectors from L2 S. There is a simple functional realization for n = 1 (see Example 3.1.6). In the general case, the right sides of the relations (1.5) express (Ff)(y) and (F−1 f)(y) as long as f ∈ L2 ∩ L1 . To check this assume first that supp f ⊂ J, where J is a bounded interval in Rn and consider a sequence {fk} approximating f ac- cording to Problem 47d. By the Hölder inequality f−fk 1 ≤ µ(J)1/2 f−fk 2, so f−fk 1 → 0, and consequently, the functions F0fk converge uniformly to Ff. On the other hand F0fk = Ffk, so F0fk − Ff 2 = fk −f 2 → 0; the sought expression of Ff then follows from the result mentioned in the notes to Section 1.2. A similar procedure can be used for a general f ∈ L2 ∩ L1 : one approximates it, e.g., by the functions fj := fχj, where χj are characteristic functions of the balls { x ∈ Rn : |x| ≤ j }. For the remaining vectors, f ∈ L2 L1 , the right side of (1.5) no longer makes sense and Ff must be defined as a limit, e.g., Ff − Ffj 2 → 0, where fj are the truncated functions defined above. The last relation is often written as (Ff)(y) = l.i.m. j→∞(2π)−n/2 |x|≤j e−i x·y f(x) dx , where the symbol l.i.m. (limes in medio) means convergence with respect to the norm of L2 (Rn ). A particular role is played by operators that map a given normed space into C. We call B(V, C) the dual space to V and denote it as V ∗ ; its elements are
- 43. 20 1 Some notions from functional analysis bounded linear functionals. Comparing it with the algebraic dual space defined in the notes, we see that V ∗ is a subspace of V f , and moreover, a Banach space with respect to the operator norm; the two dual spaces do not coincide unless dim V ∞ (Problem 54). The Hahn–Banach theorem has some simple implications. 1.5.7 Proposition: Let V0 be a subspace of a normed space V ; then (a) For any f0 ∈ V ∗ 0 there is an f ∈ V ∗ such that f | V0 = f0 and f0 = f (b) If V0 = V , then to any z ∈ V0 one can find fz ∈ V ∗ with fz = 1 such that fz | V0 = 0 and fz(z) = d(z) := infy∈V0 z−y Proof: The first assertion follows from Theorem 1.1.3 with p = f0 · . To prove (b), we have to check first that x → d(x) is a seminorm on V and d(x) = 0 holds iff x ∈ V0. Then we take the subspace V1 := { x = y+αz : y ∈ V0, α ∈ C } and set f1(x) := αd(z), in particular, f1(z) = d(z). This is obviously a linear functional and |f1(y+αz)| = d(αz) ≤ y+αz , so f1 ∈ V ∗ 1 and f1 ≤ 1. On the other hand, we have d(z) = |f1(y−z)| ≤ f1 y−z ; hence f1 = 1 and the sought functional fz is obtained by extending f1 to the whole V in accordance with the already proven part (a). 1.5.8 Corollary: (a) To any nonzero x ∈ V there is a functional fx ∈ V ∗ such that fx(x) = x and fx = 1. (b) The family V ∗ separates points of V . (c) If the dual space X∗ to a Banach space X is separable, then X is also separable. Proof: The first two assertions follow immediately from Proposition 1.5.7. Let further {fn : n = 1, 2, . . . } be a dense set in X∗ . To any nonzero fn we can find a unit vector xn ∈ X such that |fn(xn)| 1 2 fn . In view of Lemma 1.5.2 it is sufficient to check that M := {xn : n = 1, 2, . . . } is total in X. Let us assume that V0 := Mlin = X; then Proposition 1.5.7 implies the existence of a functional f ∈ X∗ such that f = 1 and f(x) = 0 for x ∈ V0. For any ε 0 we can find a nonzero fn such that fn−f ε, i.e., fn 1−ε; hence we arrive at the contradictory conclusion ε fn − f ≥ |fn(xn) − f(xn)| = |fn(xn)| 1 2 fn 1 2 (1 − ε). One of the basic problems in the Banach–space theory is to describe fully X∗ for a given X. We limit ourselves to one example; more information can be found in the notes and in the next chapter, where we shall show how the problem simplifies when X is a Hilbert space. 1.5.9 Example: The dual (p )∗ , p ≥ 1, is linearly isomorphic to q , where q := p/(p−1) for p 1 and q := ∞ for p = 1. To demonstrate this, we define fY (X) := ∞ k=1 ξkηk
- 44. 1.5 Banach spaces and operators on them 21 for any sequences X := {ξk} ∈ p and Y := {ηk} ∈ q ; then the Hölder inequality implies that fY is a bounded linear functional on p and fY ≤ Y q. The map Y → fY of q to (p )∗ is obviously linear; we claim that this is the sought isometry. We have to check its invertibility. We take an arbitrary f ∈ (p )∗ and set ηk := f(Ek), where Ek are the sequences introduced in Example 1.5.3b; then it follows from the continuity of f that the sequence Yf := {ηk}∞ k=1 fulfils f = fYf . To show that Yf ∈ q , consider first the case p 1. The vectors Xn := n k=1 sgn (ηk) |ηk|q−1 , where sgn z := z/|z| if z = 0 and zero otherwise, fulfil Xn p = ( n k=1 |ηk|q ) 1/p and f(Xn) = n k=1 |ηk|q , so the inequality |f(Xn)| ≤ f Xn p yields ∞ k=1 |ηk|q 1/q ≤ f , n = 1, 2, . . . . If p = 1 we have |f(En)| = |ηn| and En 1 = 1, so supn |ηn| ≤ f . Hence in both cases the sequence Yf ∈ q , and the obtained bounds to its norm in combination with the inequality fY ≤ Y q, which we proved above, yield f = Yf q. The dual space of a given V is normed, so we can define the second dual V ∗∗ := (V ∗ )∗ as well as higher dual spaces. For any x ∈ V we can define Jx ∈ V ∗ by Jx(f) := f(x). The map x → Jx is a linear isometry of V to a subspace of V ∗∗ (Problem 55); if its range is the whole V ∗∗ the space V is called reflexive. It follows from the definition that any reflexive space is automatically Banach. In view of Example 1.5.9, the spaces p are reflexive for p 1, and the same is true for Lp (M, dµ) (see the notes). On the other hand, 1 and C(K) are not reflexive, and similarly, L1 (M, dµ) is not reflexive unless the measure µ has a finite support. Below we shall need the following general property of reflexive spaces, which we present without proof. 1.5.10 Theorem: Any closed subspace of a reflexive space is reflexive. The notion of dual space extends naturally to topological vector spaces: the dual to (V, τ) consists of all continuous linear functionals on V ; in this case we often denote it alternatively as V . It allows us to define the weak topology τw on V as the weakest topology with respect to which any f ∈ V is continuous, or the V –weak topology in the terminology introduced in Example 1.2.5 ; in the next chapter we shall see that the definition is consistent with that of Example 1.4.3. We have τw ⊂ τ because each f ∈ V is by definition continuous with respect to τ, and it is easy to check that τw coincides with the topology generated by the family Pw := { pf : f ∈ V }, where pf (x) := |f(x)|. 1.5.11 Proposition: If (V, τ) is a locally convex space, then Pw is separating points of V and the space (V, τw) is also locally convex. Proof: If V is a normed space, the assertion follows from Corollary 1.5.8; for the general case see the notes. If a sequence {xn} ⊂ V converges to some x ∈ V with respect to τw, we write
- 45. 22 1 Some notions from functional analysis x = w limn→∞xn or xn w → x. Let us list some properties of weakly convergent sequences (only the case dim V =∞ is nontrivial — see Problem 56). 1.5.12 Theorem: (a) xn w → x iff f(xn) → f(x) holds for any f ∈ V ∗ . (b) Any weakly convergent sequence in a normed space is bounded. (c) If {xn} is a bounded sequence in a normed space V and g(xn) → g(x) for all g of some total set in F ⊂ V ∗ , then xn w → x. Proof: The first assertion follows directly from the definition. Further, we use the uniform boundedness principle which will be proven in the next section: if xn w → x, then the family {ψn} ⊂ V ∗∗ with ψn(f) := f(xn) fulfils the assumptions of Theorem 1.6.1, which yields ψn = xn c for some c 0. As for part (c), we have g(xn) → g(x) for all g ∈ Flin. Since F is total by assumption, for any f ∈ V ∗ , ε 0 there is a g ∈ Flin and a positive integer n(ε) such that f−g ε and |g(xn)−g(x)| ε for n n(ε); this easily yields |f(xn)−f(x)| ≤ (1+ xn + x )ε. However, the sequence {xn} is bounded, so f(xn) → f(x). 1.5.13 Example (weak convergence in p , p 1 ): In view of Examples 3b and 9, the family of the functionals fk({ξj}∞ j=1) := ξk , k = 1, 2, . . ., is total in (p )∗ . This means that a sequence {Xn} ⊂ p , Xn := {ξ (n) j }∞ j=1, converges weakly to X := {ξj}∞ j=1 ∈ p iff it is bounded and ξ (n) j → ξj for j = 1, 2, . . . . For instance, the sequence {En} of Example 3b converges weakly to zero; this illustrates that the two topologies are different because {En} is not norm–convergent. A topological vector space (V, τ) is called weakly complete if any sequence {xn} ⊂ V such that {f(xn)} is convergent for each f ∈ V converges weakly to some x ∈ V . A set M ⊂ V is weakly compact if any sequence {xn} ⊂ M contains a weakly convergent subsequence. 1.5.14 Theorem: Let X be a reflexive Banach space, then (a) X is weakly complete. (b) A set M ⊂ X is weakly compact iff it is bounded. Proof: (a) Let {xn} ⊂ X be such that {f(xn)} is convergent for each f ∈ V ∗ . The same argument as in the proof of Theorem 12 implies existence of a positive c such that the sequence {ψn} ⊂ X∗∗ , ψn(f) := f(xn), fulfils |ψn(f)| ≤ c f , n = 1, 2, . . ., for all f ∈ X∗ . The limit ψ(f) := limn→∞ ψn(x) exists by assumption, the map f → ψ(f) is linear, and the last inequality implies f ∈ X∗∗ . Since X is reflexive, there is an x ∈ X such that ψ(f) = f(x) for all f ∈ X∗ , i.e., xn w → x. (b) If M is not bounded there is a sequence {xn} ⊂ M such that xn n; then no subsequence of it can be weakly convergent. Suppose on the contrary that M is bounded and consider a sequence X := {xn} ⊂ M; it is clearly sufficient to
- 46. 1.6 The principle of uniform boundedness 23 assume that X is simple, xn = xm for n = m. In view of Theorem 10, Y := {xn}lin is a separable and reflexive Banach space, so Y∗∗ is also separable, and Corollary 8 implies that Y∗ is separable too. Let {gj : j = 1, 2, . . . } be a dense set in Y∗ . Since X is bounded the same is true for {g(xn)}; hence there is a subsequence X1 := {x (1) n } such that {g1(x (1) n )} converges. In a similar way, {g2(x (1) n )} is bounded, so we can pick a subsequence X2 := {x (2) n } ⊂ X1 such that {g2(x (2) n )} converges, etc.This procedure yields a chain of sequences, X ⊃ · · · ⊃ Xj ⊃ Xj+1 ⊃ · · ·, such that {gj(x (j) n )}∞ n=1 , j = 1, 2, . . ., are convergent. Now we set yn := x (n) n , so yn ∈ Xj for n ≥ j and {gj(yn)}∞ n=1 converges for any j; then {g(yn)}∞ n=1 is convergent for all g ∈ Y∗ due to Theorem 12c. The already proven part (a) implies the existence of y ∈ Y such that g(yn) → g(y) for any g ∈ Y∗ . Finally, we take an arbitrary f ∈ X∗ and denote gf := f | Y. Since gf ∈ Y∗ we have f(yn) = gf (yn) → gf (y) = f(y), and therefore yn w → y. 1.6 The principle of uniform boundedness Any Banach space X is a complete metric space so the Baire category theorem is valid in it (cf. Problem 27). Now we are going to use this fact to derive some important consequences for bounded operators on X. 1.6.1 Theorem (uniform boundedness principle): Let F ⊂ B(X, V1), where X is a Banach space and (V1, · 1) is a normed space. If supB∈F Bx 1 ∞ for any x ∈ X, then there is a positive c such that supB∈F B c. Proof: Since any operator B ∈ F is continuous, the sets Mn := {x ∈ X : Bx 1 ≤ n for all B ∈ F } are closed. Due to the assumption, we have X = ∞ n=1 Mn and by the Baire theorem, at least one of the sets Mn has an interior point, i.e., there is a natural number ñ, an x̃ ∈ Mñ, and an ε 0 such that all x fulfilling x−x̃ ε belong to Mñ, and therefore supB∈F Bx 1 ≤ ñ. Let y ∈ X be a unit vector. We set xy := ε 2 y; then xy +x̃ ∈ Mñ and By 1 = 2 ε Bxy 1 ≤ 2 ε ( B(xy +x̃) 1 + Bx̃ 1) ≤ 4ñ ε ; this implies B ≤ 4ñ ε for all B ∈ F. In what follows, X, Y are Banach spaces, Uε and Vε are open balls in X and Y, respectively, of the radius ε 0 centered at zero. By No we denote the interior of a set N ⊂ Y, i.e., the set of all its interior points. Any operator B ∈ B(X, Y) is continuous, so the pull–back B(−1) (G) of an open set G ⊂ Y is open in X. If B is surjective, the converse is also true. 1.6.2 Theorem (open–mapping theorem): If an operator B ∈ B(X, Y) is surjective and G ⊂ X is an open set, then the set BG is open in Y. We shall first prove a technical result.
- 47. 24 1 Some notions from functional analysis 1.6.3 Lemma: Let B ∈ B(X, Y) and ε 0. If (BUε)o = ∅ or (BUε)o = ∅; then 0 ∈ (BUη)0 or 0 ∈ (BUη)0 , respectively, holds for any η 0. Proof: Let y0 be an interior point of BUε; then there is δ 0 such that y0 + Vδ ⊂ BUε, i.e., to any y ∈ Vδ there exists a sequence {x (y) n } ⊂ Uε such that z (y) n := Bx (y) n → y + y0. In particular, z (0) n → y0, so z (y) n − z (0) n → y, and since x (y) n −x (o) n X 2ε we get Vδ ⊂ BU2ε. In view of Problem 33, cBUε = cBUε holds for any c 0. This implies that Vη ⊂ BUη , η := ηδ 2ε , so 0 is an interior point of BUη. A similar argument applies to (BUε)o = ∅. Proof of Theorem 1.6.2: To any x ∈ G we can find Uη such that x + Uη ⊂ G, i.e., Bx+BUη ⊂ BG. If there is Vδ ⊂ BUη the set BG is open; hence it is sufficient to check 0 ∈ (BUη)o for any η 0. We write X = ∞ n=1 Un; since B is surjective, we have Y = ∞ n=1 BUn and the Baire category theorem implies (BUñ)o = ∅ for some positive integer ñ. We shall prove that BUñ ⊂ BU2ñ. Due to the lemma, BUñ contains a ball Vδ, and this further implies Vδj ⊂ BUnj for j = 1, 2, . . ., where δj := δ/2j and nj := ñ/2j . Let y ∈ BUñ, so any neighborhood of y contains elements of BUñ; in particular, for the neighborhood y+Vδ1 we can find x1 ∈ Uñ such that Bx1 ∈ y+Vδ1 , and therefore also y−Bx1 ∈ Vδ1 ⊂ BUn1 . Repeating the argument we see that there is an x2 ∈ Un1 such that y−Bx1 −Bx2 ∈ Vδ2 ⊂ BUn2 etc; in this way we construct a sequence {xj} ⊂ X such that xj X 2ñ/2j and y − j k=1 Bxk Y δj . Then Theorem 1.6.1 implies the existence of limj→∞ j k=1 xj =: x ∈ X; we have y = Bx because B is continuous. Now x X ≤ ∞ k=1 xk X 2ñ, so y ∈ BU2ñ; this proves BUñ ⊂ BU2ñ. Since (BUñ)o = ∅ the set BU2ñ has an interior point; using the lemma again we find 0 ∈ (BUη)o for any η 0. Theorem 1.6.2 further implies the following often used result, the proof of which is left to the reader (Problem 58). 1.6.4 Corollary (inverse–mapping theorem): If B ∈ B(X, Y) is a bijection, then B−1 is a continuous linear operator from Y to X. In the rest of this section, T means a linear operator defined on a subspace DT of a Banach space X and mapping DT into a Banach space Y; we do not assume T to be continuous. The subspace DT is called the domain of the operator T and is alternatively denoted as D(T). The set Γ(T) := { [x, Tx] : x ∈ DT } is called the graph of the operator T ; it is a subspace in the Banach space X ⊕ Y. In general, a subspace Γ ⊂ X ⊕ Y is said to be a graph if each element [x, y] ∈ Γ is determined by its first component. Any graph Γ determines the linear operator TΓ from X to Y with the domain D(TΓ) := { x ∈ X : [x, y] ∈ Γ } by TΓx := y for each [x, y] ∈ Γ. It is clear that
- 48. 1.7 Spectra of closed linear operators 25 Γ(TΓ) = Γ and conversely, a linear operator is uniquely determined by its graph, TΓ(T) = T. An operator T is called closed if Γ(T) is a closed set in X ⊕ Y; the definition of the direct product gives the following equivalent expression. 1.6.5 Proposition: An operator T is closed iff for any sequence {xn} ⊂ DT such that xn → x and Txn → y, we have x ∈ DT and y = Tx. Any bounded operator is obviously closed. On the other hand, there are closed operators which are not bounded (Problem 59). If Γ(T) is not closed but its closure in X ⊕ Y is a graph (which may not be true) the operator T is said to be closable and the closed operator T := TΓ(T) is called the closure of the operator T; we have Γ(T) = Γ(T). Since Γ(T) is the smallest closed set containing Γ(T), the closure is the smallest closed extension of the operator T. Moreover, Γ(T) is a subspace in X ⊕ Y, so it is a graph iff [0, y] ∈ Γ(T) implies y = 0. This property makes it possible to describe the closure sequentially. 1.6.6 Proposition: (a) An operator T is closable iff any sequence {xn} ⊂ DT such that xn → 0 and Txn → y fulfils y = 0. (b) A vector x belongs to D(T) iff T is closable and there is a sequence {xn} ⊂ DT such that xn → x and {Txn} is convergent; if this is true, we have Txn → Tx. We have mentioned that a closed operator may not be continuous. However, this can happen only if the operator is not defined on the whole space X. 1.6.7 Theorem (closed–graph theorem): A closed linear operator T : X → Y with DT = X is continuous. Proof: Γ(T) is by assumption a closed subspace in X ⊕ Y, so it is a Banach space with the direct–sum norm, [x, y] ⊕ = x X + y Y. The map S1 : S1([x, Tx]) = x is a continuous linear bijection from Γ(T) to X, and therefore S−1 1 is continuous due to Corollary 1.6.4. The map S2 : S2([x, Tx]) = Tx is again continuous, and the same is true for the composite map S2◦S−1 1 = T. 1.7 Spectra of closed linear operators We denote by C(X) the set of all closed linear operators from a Banach space X to itself; since such operators are not necessarily bounded, one has to pay atten- tion to their domains. A complex number λ is called an eigenvalue of an operator T ∈ C(X) if T −λI is not injective, i.e., if there is a nonzero vector x ∈ DT such that Tx = λx. Any vector with this property is called an eigenvector of T (corresponding to the eigenvalue λ ). The subspace Ker (T −λ) is the respective
- 49. 26 1 Some notions from functional analysis eigenspace of T and its dimension is the multiplicity of the eigenvalue λ; in partic- ular, the latter is simple if dim Ker (T −λ) = 1. Here and in the following we use T −λ as a shorthand for T −λI where I is the unit operator on X. A subspace L ⊂ X is called T–invariant if Tx ∈ L holds for all x ∈ L∩DT ; we see that any eigenspace of T is T–invariant. Furthermore, Proposition 1.6.5 gives the following simple result. 1.7.1 Proposition: Any eigenspace of an operator T ∈ C(X) is closed. Let us now ask under which conditions the equation (T−λ)x = y can be solved for a given y ∈ X, λ ∈ C. Recall first the situation when X := V is a finite–dimensional vector space. We know that the equation with y = 0 has then a nontrivial solution if λ belongs to the spectrum of T which is defined as the set of all eigenvalues, σ(T) := { λ ∈ C : there is a nonzero x ∈ V , Tx = λx } . Since λ ∈ σ(T) holds iff det(T −λ) = 0, to find the spectrum is a purely algebraic problem; σ(T) is a nonempty set having at most dim V elements. On the other hand, the equation has a unique solution if λ ∈ σ(T) and this solution depends continuously on y. In that case therefore T −λ is a bijection of V to itself, and we get an alternative way to describe the spectrum, σ(T) = C ρ(T) , ρ(T) := { λ ∈ C : (T −λ)−1 ∈ B(V ) } . In an infinite–dimensional Banach space the two expressions are no longer equivalent: the inverse (T −λ)−1 exists if λ is not an eigenvalue of T, but it may be neither bounded nor defined on the whole X. The definition can be then formulated with the help of the second expression. Taking into account that, for T ∈ C(X) such that T −λ is invertible, the conditions (T −λ)−1 ∈ B(X) and Ran (T −λ) = X are equivalent by the closed–graph theorem and Problem 60b, we can define the spectrum of T ∈ C(X) as the set of all complex numbers λ for which T−λ is not a bijection of DT onto X. This may happen in two (mutually exclusive) cases: (i) T −λ is not injective, i.e., λ is an eigenvalue of T. (ii) T −λ is injective but Ran (T −λ) = X. The set of all eigenvalues of T forms its point spectrum σp(T). The remaining part of the spectrum which corresponds to case (ii) is divided as follows: the continuous spectrum σc(T) consists of those λ such that Ran (T −λ) is dense in X, while the points where Ran (T −λ) = X form the residual spectrum σr(T) of T. In this way, the spectrum decomposes into three disjoint sets, σ(T) = σp(T) ∪ σc(T) ∪ σr(T) . (1.6) The set ρ(T) := C σ(T) = { λ ∈ C : (T −λ)−1 ∈ B(X) }
- 50. 1.7 Spectra of closed linear operators 27 is called the resolvent set of the operator T and its elements are regular values. The map RT : ρ(T) → B(X) defined by RT (λ) := (T −λ)−1 is called the resolvent of the operator T; the same name is also often used for its values, i.e., the operators RT (λ). The starting point for derivation of basic properties of the spectrum is the following result, which is an operator analogy of the geometric–series sum. 1.7.2 Lemma: Let B ∈ B(X) and I−B 1; then B is invertible, B−1 ∈ B(X), and B−1 = lim n→∞ n j=0 (I−B)j =: ∞ j=0 (I−B)j . Proof: Denote Sn := n j=0 (I−B)j .Theinequality(1.4)gives Sn−Sm ≤ n j=m+1 I− B j for any positive integers m and n m, so {Sn} is Cauchy, and since B(X) is complete, it converges to some S ∈ B(X). We have BSn = SnB = I +Sn−Sn+1; passing to the limit we get BS = SB = I, i.e., S = B−1 . 1.7.3 Theorem: The resolvent set of an operator T ∈ C(X) is open, containing to- gether with any λ0 also its neighborhood U(λ0) := { λ ∈ C : |λ−λ0| RT (λ0) −1 }, and the resolvent is given by RT (λ) = ∞ j=0 RT (λ0)j+1 (λ−λ0)j for each λ ∈ U(λ0). Proof: The operator Bλ := I −RT (λ0)(λ−λ0) obeys the assumptions of the lemma if λ ∈ U(λ0), then B−1 λ exists and belongs to B(X). Using the obvious identities (T− λ)RT (λ0) = Bλ and RT (λ0)(T−λ)x = Bλx for all x ∈ DT , we can check that (T− λ)−1 exists and (T−λ)−1 = RT (λ0)B−1 λ ∈ B(X); hence λ ∈ ρ(T) and RT (λ0)B−1 λ = RT (λ). To finish the proof, we have to substitute for B−1 λ the expansion from the above lemma. 1.7.4 Corollary: σ(T) is a closed set. To formulate the next assertion, we need one more notion. A vector–valued function f : C → V , where V is a locally convex space, is said to be analytic in a region G ⊂ C if the derivative f (λ0) := lim λ→λ0 f(λ) − f(λ0) λ − λ0 exists for any λ0 ∈ G. One can extend to such functions some standard results of the theory of analytic functions such as the Liouville theorem; if V is, in addition, a Banach space the generalized Cauchy theorem and Cauchy integral formula are also valid — see the notes for more information. 1.7.5 Theorem: Let B be a bounded operator on a Banach space X; then the resolvent RB : ρ(B) → B(X) is an analytic function, the spectrum σ(B) is a nonempty compact set, and r(B) := sup{ |λ| : λ ∈ σ(T) } = limn→∞ Bn 1/n .
- 51. 28 1 Some notions from functional analysis Proof: By Problem 62, limλ→λ0 (RB(λ)−RB(λ0))(λ−λ0)−1 = RB(λ0)2 holds for any λ0 ∈ ρ(B), so the resolvent is analytic in ρ(B). It is analytic also at the point λ = ∞. To check this we set µ := λ−1 ; then (I−µB)−1 exists due to Lemma 1.7.2 provided |µ| B −1 . The set { λ : |λ| B } is thus contained in ρ(B), so the spectrum is bounded and therefore compact in view of Corollary 1.7.4. The obvious identity RB(λ) = −µ(I−µB)−1 yields d dµ RB(µ−1 ) µ=0 = − lim µ→0 (I−µB)−1 = −I; now we may use the generalized Liouville theorem mentioned above according to which a vector–valued function is analytic in the extended complex plane iff it is constant. Hence the assumption σ(B) = ∅ implies RB(λ) = C, i.e., (B−λ)C = I for some C ∈ B(X) and all λ ∈ C, which is impossible, however. The above argument shows at the same time that r(B) ≤ B . Applying Lemma 1.7.2 to the operator I−µB we get an alternative expression for the resol- vent, RB(λ) = −λ−1 I − ∞ k=1 λ−(k+1) Bk , (1.7) where the series converges with respect to the norm in B(X) if the numerical se- ries k Bk |λ|−k is convergent; by the Cauchy–Hadamard criterion, its radius of convergence is r0(B) := limsup n→∞ Bn 1/n . Now we set n = jk +m, where 0 ≤ m j, then Bn 1/n ≤ Bj k/n B m/n . Choosing an arbitrary fixed j and performing the limit k → ∞, we get limsup n→∞ Bn 1/n ≤ Bj 1/j , and therefore limsup n→∞ Bn 1/n ≤ infn Bn 1/n ≤ liminf n→∞ Bn 1/n ; this means that the se- quence { Bn 1/n } converges to r0(B). Since the resolvent exists for |λ| r0(B), we have r(B) ≤ r0(B) ≤ B ; the second inequality follows from the above estimate with j = 1. To finish the proof, we use the Cauchy integral formula according to which the coefficients of the series (1.7) equal Bk = − 1 2πi Cr zk RB(z) dz = − rk+1 2π 2π 0 ei(k+1)θ RB(r eiθ ) dθ , where r r(B). The circle Cr is a compact subset of ρ(B), so the function RB(·) is continuous and bounded on it by the already proved analyticity and Corollary 1.3.7b, i.e., Mr := maxz∈Cr RB(z) ∞. This yields the estimate Bk ≤ Mr rk+1 , which shows that r0(B) ≤ r for any r r0(B), and therefore r0(B) ≤ r(B). The number r(B) is called the spectral radius of the operator B; we have proven that it does not exceed the operator norm, r(B) ≤ B . Boundedness of the spectrum means that the resolvent set ρ(B) is nonempty for a bounded B. On the other hand, it may happen that ρ(T) = ∅ or σ(T) = ∅ if T ∈ C(X) B(X).
- 52. Notes to Chapter 1 29 1.7.6 Examples: (a) Consider the operators T and T(0) := T| D(0) of Problem 59. The function fλ : fλ(x) = eλx belongs to DT and Tfλ = λfλ, so σ(T) = σp(T) = C and ρ(T) = ∅. On the other hand, for a given λ ∈ C we define the operator Sλ : (Sλg)(x) = x 0 eλ(x−t) g(t) dt which maps C[0, 1] to D(0) . We have (Sλg) − λ(Sλg) = g for any g ∈ C[0, 1] and Sλ(f − λf) = f for all f ∈ D(0) , and therefore Sλ = (T(0) −λ)−1 ; in view of the closed–graph theorem and Problem 60b, Sλ is bounded, so we get ρ(T(0) ) = C and σ(T(0) ) = ∅. (b) The same conclusions are valid for the operator P̃ : P̃f := −if on L2 (0, 1) whose domain D(P̃) := AC[0, 1] consists of all functions that are absolutely continuous on the interval [0, 1] with the derivatives in L2 (0, 1), and its re- striction P(0) := P | {f ∈ AC[0, 1] : f(0) = 0 }. However, one has to use a different method to check that P̃ and P(0) are closed; we postpone the proof to Example 4.2.5. Notes to Chapter 1 Section 1.1 A linearly independent set B ⊂ V such that Blin = V is called a Hamel basis of the space V and its cardinality is the algebraic dimension of V . Such a basis exists in any vector space (Problem 4). We are more interested, however, in other bases which allow us to express elements of a space as “infinite linear combinations” of the basis vectors. This requires a topology; we shall return to that problem in the next chapter. A set C ⊂ V is convex if together with any two points it contains the line segment connecting them. Any subspace L ⊂ V is convex, and the intersection of any family of convex sets is again convex. If a point x ∈ C does not belong to the line segment connecting some y, z ∈ C different from x, it is called extremal. Equivalently, x ∈ C is an extremal point of C if x = ty + (1 − t)z with t ∈ (0, 1), y, z ∈ C implies x = y = z. The Hahn–Banach theorem is proven in most functional–analysis textbooks — see, e.g., [[ KF ]], [[ RS 1 ]], [[ Tay ]]. The set of all linear functionals on a given V becomes a vector space if we equip it with the operations defined by (αf + g)(x) := αf(x) + g(x); we call it the algebraic dual space of V and denote as V f . If V is finite–dimensional, one can check easily that V and V f are isomorphic. We often need to know whether a given family F of maps F : X → Y is “large enough” to contain for any pair of different elements x, y ∈ X a map f such that f(x) = f(y); if this is true we say that F separates points. If X, Y are vector spaces and the maps in F are linear, the family separates points if for any non–zero x ∈ X there is fx ∈ F such that fx(x) = 0. This is true, in particular, for F = Xf (Problem 7). Section 1.2 As a by–product of the argument of Example 1.2.1 we have obtained the following result: if a sequence {fn} ⊂ Lp(M, dµ) fulfils fn −f p → 0, then there is a subsequence {fnk } such that fnk (x) → f(x) for µ–a.a. x ∈ M. In fact, this is true for any positive measure and the space Lp(M, dµ) is still complete in this case — see, e.g., [[ Jar 2 ]], Thm.68; [[ DS ]], Sec.III.6. An alternative way to introduce the notion of a topology is through axiomatization of properties of the closure — see Problem 28 and [[ Kel ]] for more details. The described construction of the topological product extends easily to any finite number of topological
- 53. 30 1 Some notions from functional analysis spaces; for further generalizations see, e.g., [[Nai 1]], Sec.I.2.12. A discussion of the count- ability and separability axioms and their relations can be found in topology monographs such as [[ Al ]] or [[ Kel ]], and also in most functional–analysis textbooks. For instance, any second countable regular space is normal; the fundamental result of Uryson claims that a second countable space is normal iff its topology is induced by a metric, thus giving a partial answer to the metrizability problem. A topological space (X, τ) is connected if it cannot be written as a union of two non–empty disjoint open sets; this is equivalent to the requirement that X and ∅ are the only two sets which are simultaneously closed and open. A set M ⊂ X is connected if the space (M, τM ) with the induced topology is connected. A continuous map ϕ : [0, 1] → X is called a curve connecting the points ϕ(0) and ϕ(1). A topological space (or its subset) whose any two points can be connected by a curve is said to be arcwise (or linearly) connected. Such spaces are connected but the converse is not true. Section 1.3 There is a standard procedure called one–point compactification which allows us to construct for a given noncompact (X, τ) a compact space (X, τ) such that (X, τ) is its subspace and X X ≡ {x0} is a one–point set; the topology τ consists of all sets {x0}∪(X F), where F is a closed compact set in X — for more details see, e.g., [[Tay]], Sec.2.31. A simple example is the compactification of C by adding to it the point ∞. Any compact space is by definition σ–compact but the converse is not true — cf. [[KF]], Sec.IV.6.4. An infinite set in a σ–compact space has again at least one accumulation point; this is clear from the proof of Proposition 1.3.1. In some cases the notions of compactness and σ–compactness coincide (cf. Corollary 1.3.5 and Problem 37c). A net {yβ}β∈J is called a subnet of a net {xα}α∈I if there is a map ϕ : J → I such that (i) yβ = xϕ(β) for all β ∈ J, (ii) for any α ∈ J there is a β ∈ J such that β β implies ϕ(β) α. Using this definition, we can state the Bolzano–Weierstrass theorem: A topological space is compact iff any net in X has a convergent subnet. A proof can be found, e.g., in [[ RS 1 ]], Sec.IV.3. A map f of a metric space (X, ) to (X, ) is uniformly continuous if to any ε 0 there is δ 0 such that (f(x1), f(x2)) ε holds for any pair of points x1, x2 ∈ X fulfilling (x1, x2) ε. One can easily prove the following useful result: a continuous map of a compact space (X, ) to (X, ) is uniformly continuous. Section 1.4 A converse to Theorem 1.4.1 was proved by F. Riesz: any locally compact topological vector space is finite–dimensional — see, e.g., [[ Tay ]], Sec.3.3. Some metric– space notions do not extend directly to topological spaces but can be used after a suitable generalization. For instance, a set M in a topological vector space is bounded if to any neighborhood U of zero there is an α ∈ C such that M ⊂ αU. In view of Problem 42b, there is a positive b such that M ⊂ βU holds for all |β| ≥ b. It is easy to see that in a normed space this definition is equivalent to the requirement of existence of a cM 0 such that x cM for all x ∈ M. A locally convex space is often defined as a topological vector space in which any neighborhood of the point 0 contains a convex neighborhood of zero. We have seen that this is true for (V, τP). On the other hand, if a topological vector space (V, τ) has the stated property, there is a family of seminorms on V which separate points and generate the topology τ — see, e.g., [[ Tay ]], Sec.3.8 — so the two definitions are really equivalent. A net {xα}α∈I in a topological vector space V is Cauchy if for any neighborhood U of 0 there is a γ ∈ I such that xα−xβ ∈ U holds for all α γ , β γ. In particular, in
- 54. Notes to Chapter 1 31 a locally convex space with a topology generated by a family P the condition reads: for any ε 0 and p ∈ P, there is a γ ∈ I such that p(xα −xβ) ε for all α γ , β γ. The space V is complete if any Cauchy net in it converges to some x ∈ V . In a similar way, we define a Cauchy net in a metric space. Since such a net in a complete metric space is convergent — see, e.g., [[ DS 1 ]], Lemma I.7.5 — a locally convex space with a topology generated by a countable family P is Fréchet iff it is complete in the sense of the above definition. The completeness of S(Rn) is proved, e.g., in [[ RS 1 ]], Sec.V.3. Section 1.5 Combining the results of Example 1.5.3c with the inclusion C∞ 0 (Rn) ⊂ C(Rn) ∩ Lp(Rn, dµ) we find that the set C(Rn) ∩ Lp(Rn, dµ) is also dense in Lp(Rn, dµ). This can be proved directly for a wider class of Lp spaces — cf. [[ KF ]], Sec.VII.1.2. The proof of density of C∞ 0 (Rn) in C∞(Rn) can be found, e.g., in [[ Yo ]], Sec.1.1. In a similar way, the set C∞ 0 (Ω) for an open connected set Ω ⊂ Rn, which we shall mention below, is dense in L2(Ω) — cf. [[ RS 4 ]], Sec.XIII.14. The map F : L1(Rn) → C∞(Rn) is injective as can checked easily: the relation Fg = 0 for g ∈ L1 implies Rn f(x)g(x) dx = 0 for all f ∈ S(Rn); then J g(x) dx = 0 holds by Problem 47 for any bounded interval J ⊂ Rn and therefore also for any Borel set in Rn; this in turn means g = 0. On the other hand, F is not surjective — see, e.g., [[ KF ]], Sec.VIII.4.7; [[ Jar 2 ]], Sec.XIII.11. The product of functions f, g ∈ S(Rn) belongs to S(Rn). The relation from Problem 52 implies that its Fourier transform is fg = (2π)−n/2 ˆ f ∗ ĝ, where (f ∗ g)(x) := Rn f(x−y)g(y) dy = Rn g(x−y)f(y) dy . The map [f, g] → f ∗ g is called the convolution. It is a binary operation on S(Rn) which is obviously bilinear and commutative; the relation f ∗g = (2π)n/2 ˇ fǧ shows that it is also associative. Some important extensions of the convolution are discussed in [[Yo]], Sec.VI.3; [[ RS 2 ]], Sec.IX.1. While ( 1)∗ is isomorphic to ∞, the dual ( ∞)∗ is not isomorphic to 1. This follows from Corollary 1.5.8c because 1 is separable but ∞ is nonseparable due to Problem 18. The situation with the spaces Lp(X, dµ) is similar: (Lp)∗ , p ≥ 1, is linearly isomorphic to Lq by fϕ : fϕ(ψ) = X ϕψ dµ — see [[ Ru 1 ]], Sec.6.16; [[ Tay ]], Sec.7.4. The spaces (L∞)∗ and L1 are again nonisomorphic with the exception of the trivial case when the measure µ has a finite support; the expression of (L∞)∗ is given in [[ DS 1 ]], Sec.IV.8. These results have implications for the reflexivity of the considered spaces as mentioned in the text. The proof of Theorem 1.5.10 can be found in [[ DS 1 ]], Sec.II.3.23. Another example of Banach spaces in which one can find a general form of a bounded linear functional is represented by the spaces C(K) and CR(K) of continuous functions (complex or real–valued, respectively) on a compact Hausdorff space K. The Riesz–Markov theorem associates the functionals with Borel measures on K. In the particular case of X ≡ CR[a, b] any such measure corresponds to a function F of a bounded variation; the theorem then claims that for any f ∈ X∗ there is F of a bounded variation such that f(ϕ) = [a,b] ϕ dF holds for all ϕ ∈ X, and moreover, f is equal to the total variation of F — for details see, e.g., [[ KF ]], Sec.VI.6.6 or [[ RN ]], Sec.50. The general formulation and proof of the Riesz–Markov theorem can be found in [[ DS 1 ]], Sec.IV.6. One of the most important examples of duals to topological vector spaces is the space S(Rn) whose elements are called tempered distributions, and the dual of the complete
- 55. Exploring the Variety of Random Documents with Different Content
- 56. Mrs. de Trappe. You seem absent-minded, my dear Edith. [Pause.] I must be going now. Where are Arthur and James? We have not a moment to lose. We are going to choose wedding presents. James is going to choose Arthur's and Arthur is going to choose James's, so there can be no jealousy. It was I who thought of that way out of the difficulty. One does one's best to be nice to them, and then something happens and upsets all one's plans. Where is Cyril? Lady Dol. I am afraid Cyril is not at home. Mrs. de Trappe. Then I shall not see him. Tell him I am angry, and give my love to Julia. I hope she does not disturb you when you are in the drawing-room and have visitors. So difficult to keep a grown- up girl out of the drawing-room. Where can those men be? [Enter Lord Doldrummond, Mr. Featherleigh, and Mr. Banish.] Ah! here they are. Now, come along; we haven't a moment to lose. Good-bye, Edith. [Exeunt (after wishing their adieux) Mrs. de Trappe, Mr. Featherleigh, and Mr. Banish, Lord Doldrummond following them.] Lady Dol. [Stands alone in the middle of the room, repeating.] Cyril and—Sarah Sparrow! My son and Sarah Sparrow! And he has met her through the one woman for whom I have been wrong enough to forget my prejudices. What a punishment! [Julia enters cautiously. She is so unusually beautiful that she barely escapes the terrible charge of sublimity. But there is a certain peevishness in her expression which adds a comfortable smack of human nature to her classic features.] Julia. I thought mamma would never go. I have been hiding in your boudoir ever since I heard she was here. Lady Dol. Was Cyril with you? Julia. Oh, no; he has gone out for a walk.
- 57. Lady Dol. Tell me, dearest, have you and Cyril had any disagreement lately? Is there any misunderstanding? Julia. Oh, no. [Sighs.] Lady Dol. I remember quite well that before I married Herbert he often suffered from the oddest moods of depression. Several times he entreated me to break off the engagement. His affection was so reverential that he feared he was not worthy of me. I assure you I had the greatest difficulty in overcoming his scruples, and persuading him that whatever his faults were I could help him to subdue them. Julia. But Cyril and I are not engaged. It is all so uncertain, so humiliating. Lady Dol. Men take these things for granted. If the truth were known, I daresay he already regards you as his wife. Julia. [With an inspired air.] Perhaps that is why he treats me so unkindly. I have often thought that if he were my husband he could not be more disagreeable! He has not a word for me when I speak to him. He does not hear. Oh, Lady Doldrummond, I know what is the matter. He is in love, but I am not the one. You are all wrong. Lady Dol. No, no, no. He loves you; I am sure of it. Only be patient with him and it will come all right. Hush! is that his step? Stay here, darling, and I will go into my room and write letters. [Exit, brushing the tears from her eyes.] [Butler ushers in Mr. Mandeville. Neither of them perceive Julia, who has gone to the window.] Butler. His Lordship will be down in half an hour, sir. He is now having his hair brushed. Julia. [In surprise as she looks round.] Mr. Mandeville! [Pause.] I hardly expected to meet you here. Mandeville. And why, may I ask?
- 58. Julia. You know what Lady Doldrummond is. How did you overcome her scruples? Mandeville. Is my reputation then so very bad? Julia. You—you are supposed to be rather dangerous. You sing on the stage, and have a tenor voice. Mandeville. Is that enough to make a man dangerous? Julia. How can I tell? But mamma said you were invincible. You admire mamma, of course. [Sighs.] Mandeville. A charming woman, Mrs. de Trappe. A very interesting woman; so sympathetic. Julia. But she said she would not listen to you .Mandeville. Did she say that? [A slight pause.] I hope you will not be angry when I own that I do not especially admire your mother. A quarter of a century ago she may have had considerable attractions, but—are you offended? Julia. Offended? Oh, no. Only it seems strange. I thought that all men admired mamma. [Pause.] You have not told me yet how you made Lady Doldrummond's acquaintance. Mandeville. I am here at Lord Aprile's invitation. He has decided that he feels no further need of Lady Doldrummond's apron-strings. Julia. Oh, Mr. Mandeville, are you teaching him to be wicked? Mandeville. But you will agree with me that a young man cannot make his mother a kind of scribbling diary? Julia. Still, if he spends his time well, there does not seem to be any reason why he should refuse to say where he dines when he is not at home. Mandeville. Lady Doldrummond holds such peculiar ideas; she would find immorality in a sofa-cushion. If she were to know that Cyril is
- 59. coming with me to the dress rehearsal of our new piece! Julia. It would break her heart. And Lord Doldrummond would be indignant. Mamma says his own morals are so excellent! Mandeville. Is he an invalid? Julia. Certainly not. Why do you ask? Mandeville. Whenever I hear of a charming husband I always think that he must be an invalid. But as for morals, there can be no harm in taking Cyril to a dress rehearsal. If you do not wish him to go, however, I can easily say that the manager does not care to have strangers present. [Pause.] Afterwards there is to be a ball at Miss Sparrow's. Julia. Is Cyril going there, too? Mandeville. I believe that he has an invitation, but I will persuade him to refuse it, if you would prefer him to remain at home. Julia. You are very kind, Mr. Mandeville, but it is a matter of indifference to me where Lord Aprile goes. Mandeville. Perhaps I ought not to have mentioned this to you? Julia. [Annoyed.] It does not make the least difference. In fact, I am delighted to think that you are taking Cyril out into the world. He is wretched in this house. [With heroism.] I am glad to think that he knows anyone so interesting and clever and beautiful as Sarah Sparrow. I suppose she would be considered beautiful? Mandeville. [With a profound glance.] One can forget her— sometimes. Julia. [Looking down.] Perhaps—when I am as old as she is—I shall be prettier than I am at present. Mandeville. You always said you liked my voice. We never see anything of each other now. I once thought that—well—that you
- 60. might like me better. Are you sure you are not angry with me because I am taking Cyril to this rehearsal? Julia. Quite sure. Why should I care where Cyril goes? I only wish that I, too, might go to the theatre to-night. What part do you play? And what do you sing? A serenade? Mandeville. [Astounded.] Yes. How on earth did you guess that? The costume is, of course, picturesque, and that is the great thing in an opera. A few men can sing—after a fashion—but to find the right clothes to sing in—that shows the true artist. Julia. And Sarah; does she look her part? Mandeville. Well, I do not like to say anything against her, but she is not quite the person I should cast for la Marquise de la Perdrigonde. Ah! if you were on the stage, Miss de Trappe! You have just the exquisite charm, the grace, the majesty of bearing which, in the opinion of those who have never been to Court, is the peculiar distinction of women accustomed to the highest society. Julia. Oh, I should like to be an actress! Mandeville. No! no! I spoke selfishly—if you only acted with me, it would be different; but—but I could not bear to see another man making love to you—another man holding your hand and singing into your eyes—and—and——Oh, this is madness. You must not listen to me. Julia. I am not—angry, but—you must never again say things which you do not mean. If I thought you were untruthful it would make me so—so miserable. Always tell me the truth. [Holds out her hand.] Mandeville. You are very beautiful! [She drops her eyes, smiles, and wanders unconsciously to the mirror.]
- 61. [Lady Doldrummond suddenly enters from the boudoir, and Cyril from the middle door. Cyril is handsome, but his features have that delicacy and his expression that pensiveness which promise artistic longings and domestic disappointment.] Cyril. [Cordially and in a state of suppressed excitement.] Oh, mother, this is my friend Mandeville. You have heard me mention him? Lady Dol. I do not remember, but—— Cyril. When I promised to go out with you this afternoon, I forgot that I had another engagement. Mandeville has been kind enough to call for me. Lady Dol. Another engagement, Cyril? [Lord Doldrummond enters and comes down, anxiously looking from one to the other.] Cyril. Father, this is my friend Mandeville. We have arranged to go up to town this afternoon. Lady Dol. [Calmly.] What time shall I send the carriage to the station for you? The last train usually arrives about—— Cyril. I shall not return to-night. I intend to stay in town. Mandeville will put me up. Lord Dol. And where are you going? Mandeville. He is coming to our dress rehearsal of the “Dandy and the Dancer.” Cyril. At the Parnassus. [Lord and Lady Doldrummond exchange horrified glances.] I daresay you have never heard of the place, but it amuses me to go there, and I must learn life for myself. I am two- and-twenty, and it is not extraordinary that I should wish to be my own master. I intend to have chambers of my own in town.
- 62. Lady Dol. Surely you have every liberty in this house? Lord Dol. If you leave us, you will leave the rooms in which your mother has spent every hour of her life, since the day you were born, planning and improving. Must all her care and thought go for nothing? The silk hangings in your bedroom she worked with her own hands. There is not so much as a pen-wiper in your quarter of the house which she did not choose with the idea of giving you one more token of her affection. Cyril. I am not ungrateful, but I cannot see much of the world through my mother's embroidery. As you say, I have every comfort here. I may gorge at your expense and snore on your pillows and bully your servants. I can do everything, in fact, but live. Dear mother, be reasonable. [Tries to kiss her. She remains quite frigid.] [Footman enters.] Footman. The dog-cart is at the door, my lord. Cyril. You think it well over and you will see that I am perfectly right. Come on, Mandeville, we shall miss the train. Make haste: there is no time to be polite. [He goes out, dragging Mandeville after him, and ignoring Julia.] Lord Dol. Was that my son? I am ashamed of him! To desert us in this rude, insolent, heartless manner. If I had whipped him more and loved him less, he would not have been leaving me to lodge with a God knows who. I disown him! The fool! Lady Dol. If you have anything to say, blame me! Cyril has the noblest heart in the world; I am the fool. Curtain.
- 63. Transcriber's Notes: Scroll the mouse over words in Greek and the transliteration will appear. Punctuation was standardized. Words in dialect, obsolete or alternative spellings were not changed. The following were corrected: missing 'f' added to of allution to allusion needed to heeded undiscouragable to undiscourageable snggest to suggest gasp to grasp deing to being geos to goes Gardi to Guardi waning to waving allign to align poem to poet requiees to requires upsettting to upsetting missing 'l' added to small Illustration Transcriptions: Cover: The Yellow Book An Illustrated Quarterly Volume I April 1894 London: Elkin Mathews John Lane
- 64. [The following text was cropped out of the illustration: Boston: Copeland Day Price 5/-] Inner Cover: The Yellow Book An Illustrated Quarterly Volume I April 1894 London: Elkin Mathews John Lane Boston: Copeland Day
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