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An Introduction to Functional Analysis
This accessible text covers key results in functional analysis that are essential for fur-
ther study in the calculus of variations, analysis, dynamical systems, and the theory of
partial differential equations. The treatment of Hilbert spaces covers the topics required
to prove the Hilbert–Schmidt Theorem, including orthonormal bases, the Riesz Repre-
sentation Theorem, and the basics of spectral theory. The material on Banach spaces
and their duals includes the Hahn–Banach Theorem, the Krein–Milman Theorem, and
results based on the Baire Category Theorem, before culminating in a proof of sequen-
tial weak compactness in reflexive spaces. Arguments are presented in detail, and more
than 200 fully-worked exercises are included to provide practice applying techniques
and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces
and point-set topology is assumed, but knowledge of measure theory is not required,
making this book ideal for upper undergraduate-level and beginning graduate-level
courses.
JA M E S RO B I N S O N is a professor in the Mathematics Institute at the University of
Warwick. He has been the recipient of a Royal Society University Research Fellowship
and an EPSRC Leadership Fellowship. He has written six books in addition to his many
publications in infinite-dimensional dynamical systems, dimension theory, and partial
differential equations.

An Introduction to Functional Analysis
JAMES C. ROBINSON
University of Warwick

University Printing House, Cambridge CB2 8BS, United Kingdom
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Information on this title: www.cambridge.org/9780521899642
DOI: 10.1017/9781139030267
c
James C. Robinson 2020
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2020
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Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.

Contents
Preface page xiii
PART I PRELIMINARIES 1
1 Vector Spaces and Bases 3
1.1 Definition of a Vector Space 3
1.2 Examples of Vector Spaces 4
1.3 Linear Subspaces 6
1.4 Spanning Sets, Linear Independence, and Bases 7
1.5 Linear Maps between Vector Spaces and Their Inverses 10
1.6 Existence of Bases and Zorn’s Lemma 13
Exercises 15
2 Metric Spaces 17
2.1 Metric Spaces 17
2.2 Open and Closed Sets 19
2.3 Continuity and Sequential Continuity 22
2.4 Interior, Closure, Density, and Separability 23
2.5 Compactness 25
Exercises 30
PART II NORMED LINEAR SPACES 33
3 Norms and Normed Spaces 35
3.1 Norms 35
3.2 Examples of Normed Spaces 38
3.3 Convergence in Normed Spaces 42
3.4 Equivalent Norms 43
3.5 Isomorphisms between Normed Spaces 46
3.6 Separability of Normed Spaces 48
Exercises 50
vii

viii Contents
4 Complete Normed Spaces 53
4.1 Banach Spaces 53
4.2 Examples of Banach Spaces 56
4.2.1 Sequence Spaces 57
4.2.2 Spaces of Functions 58
4.3 Sequences in Banach Spaces 61
4.4 The Contraction Mapping Theorem 63
Exercises 64
5 Finite-Dimensional Normed Spaces 66
5.1 Equivalence of Norms on Finite-Dimensional Spaces 66
5.2 Compactness of the Closed Unit Ball 68
Exercises 70
6 Spaces of Continuous Functions 71
6.1 The Weierstrass Approximation Theorem 71
6.2 The Stone–Weierstrass Theorem 77
6.3 The Arzelà–Ascoli Theorem 83
Exercises 86
7 Completions and the Lebesgue Spaces L p() 89
7.1 Non-completeness of C([0, 1]) with the L1 Norm 89
7.2 The Completion of a Normed Space 91
7.3 Definition of the L p Spaces as Completions 94
Exercises 97
PART III HILBERT SPACES 99
8 Hilbert Spaces 101
8.1 Inner Products 101
8.2 The Cauchy–Schwarz Inequality 103
8.3 Properties of the Induced Norms 105
8.4 Hilbert Spaces 107
Exercises 108
9 Orthonormal Sets and Orthonormal Bases for Hilbert Spaces 110
9.1 Schauder Bases in Normed Spaces 110
9.2 Orthonormal Sets 112
9.3 Convergence of Orthogonal Series 115
9.4 Orthonormal Bases for Hilbert Spaces 117
9.5 Separable Hilbert Spaces 122
Exercises 123

Contents ix
10 Closest Points and Approximation 126
10.1 Closest Points in Convex Subsets of Hilbert Spaces 126
10.2 Linear Subspaces and Orthogonal Complements 129
10.3 Best Approximations 131
Exercises 134
11 Linear Maps between Normed Spaces 137
11.1 Bounded Linear Maps 137
11.2 Some Examples of Bounded Linear Maps 141
11.3 Completeness of B(X, Y) When Y Is Complete 145
11.4 Kernel and Range 146
11.5 Inverses and Invertibility 147
Exercises 150
12 Dual Spaces and the Riesz Representation Theorem 153
12.1 The Dual Space 153
12.2 The Riesz Representation Theorem 155
Exercises 157
13 The Hilbert Adjoint of a Linear Operator 159
13.1 Existence of the Hilbert Adjoint 159
13.2 Some Examples of the Hilbert Adjoint 162
Exercises 164
14 The Spectrum of a Bounded Linear Operator 165
14.1 The Resolvent and Spectrum 165
14.2 The Spectral Mapping Theorem for Polynomials 169
Exercises 171
15 Compact Linear Operators 173
15.1 Compact Operators 173
15.2 Examples of Compact Operators 175
15.3 Two Results for Compact Operators 177
Exercises 178
16 The Hilbert–Schmidt Theorem 180
16.1 Eigenvalues of Self-Adjoint Operators 180
16.2 Eigenvalues of Compact Self-Adjoint Operators 182
16.3 The Hilbert–Schmidt Theorem 184
Exercises 188
17 Application: Sturm–Liouville Problems 190
17.1 Symmetry of L and the Wronskian 191

x Contents
17.2 The Green’s Function 193
17.3 Eigenvalues of the Sturm–Liouville Problem 195
PART IV BANACH SPACES 199
18 Dual Spaces of Banach Spaces 201
18.1 The Young and Hölder Inequalities 202
18.2 The Dual Spaces of p 204
18.3 Dual Spaces of L p() 207
Exercises 208
19 The Hahn–Banach Theorem 210
19.1 The Hahn–Banach Theorem: Real Case 210
19.2 The Hahn–Banach Theorem: Complex Case 214
Exercises 217
20 Some Applications of the Hahn–Banach Theorem 219
20.1 Existence of a Support Functional 219
20.2 The Distance Functional 220
20.3 Separability of X∗ Implies Separability of X 221
20.4 Adjoints of Linear Maps between Banach Spaces 222
20.5 Generalised Banach Limits 224
Exercises 226
21 Convex Subsets of Banach Spaces 228
21.1 The Minkowski Functional 228
21.2 Separating Convex Sets 230
21.3 Linear Functionals and Hyperplanes 233
21.4 Characterisation of Closed Convex Sets 234
21.5 The Convex Hull 235
21.6 The Krein–Milman Theorem 236
Exercises 239
22 The Principle of Uniform Boundedness 240
22.1 The Baire Category Theorem 240
22.2 The Principle of Uniform Boundedness 242
22.3 Fourier Series of Continuous Functions 244
Exercises 247
23 The Open Mapping, Inverse Mapping, and Closed Graph
Theorems 249
23.1 The Open Mapping and Inverse Mapping Theorems 249
23.2 Schauder Bases in Separable Banach Spaces 252

Contents xi
23.3 The Closed Graph Theorem 255
Exercises 256
24 Spectral Theory for Compact Operators 258
24.1 Properties of T − I When T Is Compact 258
24.2 Properties of Eigenvalues 262
25 Unbounded Operators on Hilbert Spaces 264
25.1 Adjoints of Unbounded Operators 265
25.2 Closed Operators and the Closure of Symmetric Operators 267
25.3 The Spectrum of Closed Unbounded Self-Adjoint Operators 269
26 Reflexive Spaces 273
26.1 The Second Dual 273
26.2 Some Examples of Reflexive Spaces 275
26.3 X Is Reflexive If and Only If X∗ Is Reflexive 277
Exercises 280
27 Weak and Weak-∗ Convergence 282
27.1 Weak Convergence 282
27.2 Examples of Weak Convergence in Various Spaces 285
27.2.1 Weak Convergence in p, 1 p ∞ 285
27.2.2 Weak Convergence in 1: Schur’s Theorem 286
27.2.3 Weak versus Pointwise Convergence in C([0, 1]) 288
27.3 Weak Closures 289
27.4 Weak-∗ Convergence 290
27.5 Two Weak-Compactness Theorems 292
Exercises 295
APPENDICES 299
Appendix A Zorn’s Lemma 301
Appendix B Lebesgue Integration 305
Appendix C The Banach–Alaoglu Theorem 319
Solutions to Exercises 331
References 394
Index 396

Preface
This book is intended to cover the core functional analysis syllabus and, in
particular, presents many of the results that are needed in partial differential
equations, the calculus of variations, or dynamical systems. The material is
developed far enough that the next step would be application to one of these
areas or further pursuit of ‘functional analysis’ itself at a significantly more
advanced level.
The content is based on the two functional analysis modules taught at the
University of Warwick to our third-year undergraduates. As such, it should be
straightforward to use this book (with some judicious pruning) as the basis
of a two-term course, with Part III (Hilbert spaces) taught in the first term and
Part IV (Banach spaces) in the second term. Part II contains foundational mate-
rial (a general theory of normed spaces and a collection of example spaces) that
is needed for both Parts III and IV; some of this material could find a home
in either term, according to taste. A one-term standalone module on Banach
spaces could be based on Part II; Chapters 11, 14, and 15 from Part III; and
Part IV.
Familiarity is assumed with the theory of finite-dimensional vector spaces
and basic point-set topology (metric spaces, open and closed sets, compact-
ness, and completeness), which is revised, at a fairly brisk pace and with some
proofs omitted, in the first two chapters. No knowledge of measure theory or
Lebesgue integration is required: the Lebesgue spaces are introduced as com-
pletions of the space of continuous functions in Chapter 7, with the standard
construction of the Lebesgue integral outlined in Appendix B. The canoni-
cal examples of non-Hilbert Banach spaces used in Part IV are the sequence
spaces p rather than the Lebesgue spaces L p; I hope that this will make the
book accessible to a wider audience. In the same spirit I have tried to spell
xiii

xiv Preface
out all the arguments in detail; there are no1 four-line proofs that when written
with all the details expand to fill the same number of pages.
For the most part the approach adopted here is to cover the simpler case of
Hilbert spaces in Part III before turning to Banach spaces, for which the theory
becomes more abstract, in Part IV. There is an argument that it is more effi-
cient to prove results in Banach spaces before specialising to Hilbert spaces,
but my suspicion is that this is a product of familiarity and experience: in
the same way one might argue that it is more economical to teach analysis
in metric spaces before specialising to the particular case of real sequences
and real-valued functions. That said, some basic concepts and results are not
significantly simpler in Hilbert spaces, so portions of Parts II and III deal with
Banach rather than Hilbert spaces.
By way of a very brief overview of the contents of the book, it is perhaps
useful to describe the end points of Parts III and IV. Part III works towards the
Hilbert–Schmidt Theorem that decomposes a self-adjoint compact operator on
a Hilbert space in terms of its eigenvalues and eigenfunctions, and then applies
this to the example of the Sturm–Liouville eigenvalue problem. It therefore
covers orthonormal bases, orthogonal projections, the Riesz Representation
Theorem, and the basics of spectral theory. Part IV culminates with the result
that the closed unit ball in a reflexive Banach space is weakly sequentially
compact. So this part covers dual spaces in more detail, the Hahn–Banach
Theorem and applications to convex sets, results for linear operators based on
the Baire Category Theorem, reflexivity, and weak and weak-∗ convergence.
Almost every chapter ends with a collection of exercises, and full solutions
to these are given at the end of the book.
There are three appendices. The first shows the equivalence of Zorn’s
Lemma and the Axiom of Choice; the second provides a quick overview of the
construction of the Lebesgue integral and proves properties of the Lebesgue
spaces that rely on measure-theoretic techniques; and the third proves the
Banach–Alaoglu Theorem on weak-∗ compactness of the closed unit ball in
an arbitrary Banach space, a topological result that lies outside the scope of
the main part of the book.
I am indebted to those at Warwick who taught the Functional Analysis
courses before me, both in the selection of the material and the general
approach. Although I have adapted both over the years, the skeleton of this
book was provided by Robert MacKay and Keith Ball, to whom I am very
1 Actually, there is one. An abridged version of the proof that (L p)∗ ≡ Lq appears in Chapter 18
and takes about half a page. The detailed proof, which requires some non-trivial measure theory,
takes up two pages Appendix B.

Preface xv
grateful. Those who have subsequently taught the same material, Richard
Sharp and Vassili Gelfreich, have also been extremely helpful.
Writing a textbook encourages a magpie approach to results, proofs, and
examples. I have been extremely fortunate that there are already a large num-
ber of texts on functional analysis, and I have tried to take advantage of the
many insights and the imaginative problems that they contain. Just as there are
standard results and standard proofs, there are many standard exercises, but I
have credited those that I have adopted that seemed particularly imaginative or
unusual. In addition, there is a long list of references at the back of the book,
and each of these has contributed something to this text. I would particularly
like to acknowledge the book by Rynne and Youngson (2008) and the older
texts by Kreyszig (1978) and Pryce (1973) as consistent sources of inspiration.
The books by Giles (2000) and Lax (2002) contain many interesting examples
and exercises.
I have not tried to trace the history of the many now ‘classical’ results that
occur throughout the book. For those who are interested in this aspect of the
subject, Giles (2000) has an appendix that gives a nice overview of the his-
torical background, and historical comments are woven throughout the text by
Lax (2002). Banach’s 1932 monograph contains a significant proportion of the
results in Part IV.
Many staff at Cambridge University Press have been involved with this
project over the years: Clare Dennison, Sam Harrison, Amy He, Kaitlin Leach,
Peter Thompson, and David Tranah. Given such a long list of names, it goes
without saying that I would like to thank them all for their patience and support
(and apologise to anybody I have missed). I would particularly like to thank
Kaitlin for ultimately holding me to a deadline that meant I finally finished the
book.
Lastly, I am extremely grateful to Wojciech Ożański, who read a draft ver-
sion of this book and provided me with many corrections, suggestions, and
insightful comments.

1
Vector Spaces and Bases
Much of the theory of ‘functional analysis’ that we will consider in this book is
an infinite-dimensional version of results familiar for linear operators between
finite-dimensional vector spaces. We therefore start by recalling some of the
basic theory of linear algebra, beginning with the formal definition of a vector
space. We then discuss linear maps between vector spaces, and end by proving
that every vector space has a basis using Zorn’s Lemma. Proofs of basic results
from linear algebra can be found in Friedberg et al. (2004) or in Chapter 4 of
Naylor and Sell (1982), for example.
1.1 Definition of a Vector Space
The linear spaces that occur naturally in functional analysis are vector spaces
defined over R or C; we will refer to real or complex vector spaces respectively,
but generally we will omit the word ‘real’ or ‘complex’ unless we need to make
an explicit distinction between the two cases.
Throughout the book we use the symbol K to denote either R or C.
Definition 1.1 A vector space V over K is a set V along with notions of
addition in V and multiplication by scalars, i.e.
x + y ∈ V for x, y ∈ V and λx ∈ V for λ ∈ K, x ∈ V,
(1.1)
such that
(i) additive and multiplicative identities exist: there exists a zero element
0 ∈ V such that x + 0 = x for all x ∈ V ; and 1 ∈ K is the identity
for scalar multiplication, 1x = x for all x ∈ V ;
(ii) there are additive inverses: for every x ∈ V there exists an element
−x ∈ V such that x + (−x) = 0;
3

4 Vector Spaces and Bases
(iii) addition is commutative and associative,
x + y = y + x and x + (y + z) = (x + y) + z,
for all x, y, z ∈ V ; and
(iv) multiplication is associative,
α(βx) = (αβ)x for all α, β ∈ K, x ∈ V,
and distributive,
α(x + y) = αx + αy and (α + β)x = αx + βx
for all α, β ∈ K, x, y ∈ V .
In checking that a particular collection V is a vector space over K, properties
(i)–(iv) are often immediate; one usually has to check only that V is closed
under addition and scalar multiplication (i.e. that (1.1) holds).
1.2 Examples of Vector Spaces
Of course, Rn is a real vector space over R; but is not a vector space over C,
since ix /
∈ Rn for any1 x ∈ Rn. In contrast, Cn can be a vector space over
both R and C; the space Cn over R is (according to the terminology intro-
duced above) a ‘real vector space’. This example is a useful illustration that
the real/complex label refers to the field K, i.e. the allowable scalar multiples,
rather than to the elements of the space itself.
Given any two vector spaces V1 and V2 over K, the product space V1 × V2
consisting of all pairs (v1, v2) with v1 ∈ V1 and v2 ∈ V2 is another vector
space if we define
(v1, v2)+(u1, u2) := (v1 +u1, v2 +u2) and α(v1, v2) := (αv1, αv2),
for v1, u1 ∈ V1, v2, u2 ∈ V2, α ∈ K.
We now introduce some less trivial examples.
Example 1.2 The space F(U, V ) of all functions f : U → V , where U and
V are both vector spaces over the same field K, is itself a vector space, if we
use the obvious definitions of what addition and scalar multiplication should
mean for functions. We give these definitions here for the one and only time:
1 Throughout this book we will use a bold x for elements of Rn (also of Cn), with x given in
components by x = (x1, . . . , xn).

1.2 Examples of Vector Spaces 5
for f, g ∈ F(U, V ) and α ∈ K, we denote by f + g the function from U to V
whose values are given by
( f + g)(x) = f (x) + g(x), x ∈ U,
(‘pointwise addition’) and by αf the function whose values are
(αf )(x) = α f (x), x ∈ U
(‘pointwise multiplication’).
Example 1.3 The space C([a, b]; K) of all K-valued continuous functions
on the interval [a, b] is a vector space. We will often write C([a, b]) for
C([a, b]; R).
Proof The sum of two continuous functions is again continuous, as is any
scalar multiple of a continuous function.
Example 1.4 The space P(I) of all real polynomials on any interval I ⊂ R,
P(I) =
⎧
⎨
⎩
p: I → R : p(x) =
n

j=0
aj x j
, n = 0, 1, 2, . . . , aj ∈ R
⎫
⎬
⎭
is a vector space.
The next example introduces a family of spaces that will prove to be
particularly important.
Example 1.5 For 1 ≤ p ∞ the space p(K) consists of all pth power
summable sequences x = (x j )∞
j=1 with elements in K, i.e.
p
(K) =
⎧
⎨
⎩
x = (x j )∞
j=1 : x j ∈ K,
∞

j=1
|x j |p
∞
⎫
⎬
⎭
.
For p = ∞, ∞(K) is the space of all bounded sequences in K. Sometimes we
will simply write p for p(K). Note that, as with Kn, we will use a bold x to
denote a particular sequence in p.
For x, y ∈ p(K) we set
x + y := (x1 + y1, x2 + y2, . . .),
and for α ∈ K, x ∈ p, we define
αx := (αx1, αx2, . . .).
With these definitions p(K) is a vector space.

Proof The only thing that is not immediate is whether x + y ∈ p(K) if
x, y ∈ p(K). This is clear when p = ∞, since
sup
j∈N
|x j + yj | ≤ sup
j∈N
|x j | + sup
j∈N
|yj | ∞.
For 1 ≤ p ∞ this follows using the inequality
(a + b)p
≤ [2 max(a, b)]p
≤ 2p
(ap
+ bp
), for a, b ≥ 0; (1.2)
for every n ∈ N we have
n

j=1
|x j + yj |p
≤
n

j=1
2p
(|x j |p
+ |yj |p
) ≤ 2p
∞

j=1
|x j |p
+ 2p
∞

j=1
|yj |p
∞
and so ∞
j=1 |x j + yj |p ∞ as required.
(The factor 2p in (1.2) can be improved to 2p−1; see Exercise 1.1.)
1.3 Linear Subspaces
If V is a vector space (over K) then any subset U ⊂ V is a subspace of V if U is
again a vector space, i.e. if it is closed under addition and scalar multiplication,
i.e. u1 + u2 ∈ U for every u1, u2 ∈ U and λu ∈ U for every λ ∈ K, u ∈ U.
Example 1.6 For any y ∈ Rn, the set
{x ∈ Rn
: x · y = 0}
is a subspace of Rn.
Example 1.7 The set
X = f ∈ C([−1, 1]) :
ˆ 0
−1
f (x) dx = 0,
ˆ 1
0
f (x), dx = 0
is a subspace of C([−1, 1]).
Example 1.8 The space c0(K) of all null sequences, i.e. of all sequences
x = (x j )∞
j=1 such that x j → 0 as j → ∞, is a subspace of ∞(K), and
for every 1 ≤ p ∞ the space p(K) is a subspace of c0(K).
The space c00(K) of all sequences with only a finite number of non-zero
terms is a subspace of c0(K) and of p(K) for every 1 ≤ p ≤ ∞.

Proof For the inclusion properties of c0(K), note that any convergent
sequence (in particular any null sequence) is bounded, which shows that
c0(K) ⊂ ∞(K). If x ∈ p, 1 ≤ p ∞, then ∞
j=1 |x j |p ∞, which
implies that |x j |p → 0 as j → ∞, so x ∈ c0(K). The properties of c00(K) are
immediate.
1.4 Spanning Sets, Linear Independence, and Bases
We now recall the definition of a vector-space basis, which will also allow us
to define the dimension of a vector space.
Definition 1.9 The linear span of a subset E of a vector space V is the
collection of all finite linear combinations of elements of E:
Span(E) =
⎧
⎨
⎩
v ∈ V : v =
n

j=1
αj ej , for some n ∈ N, αj ∈ K, ej ∈ E
⎫
⎬
⎭
.
We say that E spans V if V = Span(E).
If E spans V this means that we can write any v ∈ V in the form
v =
n

j=1
αj ej ,
i.e v can be expressed as a finite linear combination of elements of E. (Once
we have a way to discuss convergence we will also be able to consider ‘infinite
linear combinations’, but these are not available when we can only use the
vector-space axioms.)
Definition 1.10 A set E ⊂ V is linearly independent if any finite collection
of elements of E is linearly independent, i.e.
n

j=1
αj ej = 0 ⇒ α1 = · · · = αn = 0
for any choice of n ∈ N, αj ∈ K, and ej ∈ E.
To distinguish the standard definition of a basis for a vector space from the
notion of a ‘Schauder basis’, which we will meet later, we refer to such a basis
as a ‘Hamel basis’.

Definition 1.11 A Hamel basis for a vector space V is any linearly indepen-
dent spanning set.
Expansions in terms of basis elements are unique (for a proof see Exercise
1.3).
Lemma 1.12 If E is a Hamel basis for V, then any element of V can be written
uniquely in the form
v =
n

j=1
αj ej
for some n ∈ N, αj ∈ K, and ej ∈ E.
Any Hamel basis E of V must be a maximal linearly independent set, i.e.
E is linearly independent and E ∪ {v} is not linearly independent for any
v ∈ V E. We now show that this can be reversed.
Lemma 1.13 If E ⊂ V is maximal linearly independent set, then E is a Hamel
basis for V .
Proof To show that E is a Hamel basis we only need to show that it spans V,
since it is linearly independent by assumption.
If E does not span V, then there exists some v ∈ V that cannot be written as
any finite linear combination of the elements of E. To obtain a contradiction,
we show that in this case E ∪ {v} must be a linearly independent set. Choose
n ∈ N and {ej }n
j=1 ∈ E, and suppose that
n

j=1
αj ej + αn+1v = 0.
Since v cannot be written as a sum of any finite collection of the {ej }, we must
have αn+1 = 0, which leaves n
j=1 αj ej = 0. However, since E is linearly
independent and {ej }n
j=1 is a finite subset of E it follows that αj = 0 for all
j = 1, . . . , n. Since we already have αn+1 = 0, it follows that E ∪ {v} is
linearly independent, contradicting the fact that E is a maximal linearly
independent set. So E spans V, as claimed.
If V has a basis consisting of a finite number of elements, then every basis
of V contains the same number of elements (for a proof see Exercise 1.4).
Lemma 1.14 If V has a basis consisting of n elements, then every basis for V
has n elements.

This result allows us to make the following definition of the dimension of a
vector space.
Definition 1.15 If V has a basis consisting of a finite number of elements, then
V is finite-dimensional and the dimension of V is the number of elements in
this basis. If V has no finite basis, then V is infinite-dimensional.
Since a basis is a maximal linearly independent set (Lemma 1.13), it follows
that a space is infinite-dimensional if and only if for every n ∈ N one can find
a set of n linearly independent elements of V .
Example 1.16 For every 1 ≤ p ≤ ∞ the space p(K) is infinite-dimensional.
Proof Let us define for each j ∈ N the sequence
e( j)
= (0, 0, . . . , 1, 0, . . .), (1.3)
which consists entirely of zeros apart from having 1 as its jth term. We can
also write
e
( j)
i = δi j :=
1 i = j
0 i = j,
(1.4)
where δi j is the Kronecker delta. These are all elements of p(K) for every
p ∈ [1, ∞], and will frequently prove useful in what follows.
For any n ∈ N the n elements {e( j)}n
j=1 are linearly independent, since
n

j=1
αj e( j)
= (α1, α2, . . . , αn, 0, 0, 0, . . .) = 0
implies that α1 = α2 = · · · = αn = 0. It follows that p(K) is an infinite-
dimensional vector space.
Example 1.17 The vector space C([0, 1]; K) is infinite-dimensional.
Proof For any n ∈ N the functions {1, x, x2, . . . , xn} are linearly independent:
if
f (x) :=
n

j=0
αj x j
= 0 for every x ∈ [0, 1],

then αj = 0 for every j. To see this, first set x = 0, which shows that α0 = 0,
then differentiate once to obtain
f
(x) =
n

j=1
αj jx j−1
= 0
and set x = 0 to show that α1 = 0. Continue differentiating repeatedly, each
time setting x = 0 to show that αj = 0 for all j = 0, . . . , n.
1.5 Linear Maps between Vector Spaces and Their Inverses
Vector spaces have a linear structure, i.e. we can add elements and multiply by
scalars. When we consider maps from one vector space to another, it is natural
to consider maps that respect this linear structure.
Definition 1.18 If X and Y are vector spaces over K, then a map T : X → Y
is linear if
T (x + x
) = T (x) + T (x
) and T (αx) = αT (x), α ∈ K, x, x
∈ X.
(This is the same as requiring that T (αx + βx) = αT (x) + βT (x) for any
α, β ∈ K, x, x ∈ U.)
We often omit the brackets around the argument, and write T x for T (x)
when T is linear.
Note that the definition of what it means to be linear involves the field K. So,
for example, if we take X = Y = C and let T (z) = z (the complex conjugate
of z), this map is linear if we take K = R, but not if we take K = C. We always
have
T (z + w) = z + w = z + w = T (z) + T (w), z, w ∈ C,
but the linearity property for scalar multiples only holds if α ∈ R, since
T (αz) = αz = α z
and this is equal to αz = αT (z) if and only if α ∈ R.
This kind of ‘conjugate-linear’ behaviour is common enough that it is worth
making a formal definition.

1.5 Linear Maps between Vector Spaces and Their Inverses 11
Definition 1.19 If X and Y are vector spaces over C, then a map T : X → Y
is conjugate-linear if
T (x + x
) = T x + T x
and T (αx) = α T x, α ∈ C, x, x
∈ X.
(Such maps are sometimes called anti-linear.)
The space of all linear maps from X into Y we write as L(X, Y), and when
Y = X we abbreviate this to L(X). This is a vector space: for T1, T2 ∈ L(X, Y)
and α ∈ K we define T1 + T2 and αT1 by setting
(T1 + T2)(x) = T1x + T2x and (αT1)(x) = αT1x, x ∈ X.
With these definitions a linear combination of two linear maps is again a linear
map:
T1, T2 ∈ L(X, Y) ⇒ αT1 + βT2 ∈ L(X, Y), α, β ∈ K.
Similarly the composition of compatible linear maps is again linear,
T ∈ L(X, Y), S ∈ L(Y, Z) ⇒ S ◦ T ∈ L(X, Z)
since
(S ◦ T )(αx + βx
) = S(αT x + βT x
) = α(S ◦ T )x + β(S ◦ T )x
.
Definition 1.20 If T ∈ L(X, Y), then we define its kernel as
Ker(T ) := {x ∈ X : T x = 0}
and its range (or image) as
Range(T ) := {y ∈ Y : y = T x for some x ∈ X}.
These are both vector spaces (see Exercise 1.5).
One particularly simple (but important) example of a linear map is the
identity map IX : X → X given by IX (x) = x.
Recall that a map T : X → Y is injective (or one-to-one) if
T x = T x
⇒ x = x
.
To check if a linear map T : X → Y is injective, it is enough to show that its
kernel is trivial, i.e. that Ker(T ) = {0}.
Lemma 1.21 A map T ∈ L(X, Y) is injective if and only if Ker(T ) = {0}.

Proof We prove the equivalent statement that T is not injective if and only if
Ker(T ) = {0}.
If T is not injective, then there exist x1, x2 ∈ X with x1 = x2 such that
T x1 = T x2, i.e. T (x1 − x2) = 0, and so x1 − x2 ∈ Ker(T ) and therefore
Ker(T ) = {0}. On the contrary, if z ∈ Ker(T ) with z = 0, then for any x1 ∈ X
we have T (x1 + z) = T x1 and T is not injective.
A map T : X → Y is surjective (or onto) if for every y ∈ Y there exists
x ∈ X such that T x = y.
Lemma 1.22 If X is a finite-dimensional vector space and T ∈ L(X), then T
is injective if and only if T is surjective.
Proof The Rank–Nullity Theorem (e.g. Theorem 2.3 in Friedberg et al. (2014)
or Theorem 4.7.7 in Naylor and Sell (1982)) guarantees that
dim(Ker(T )) + dim(Range(T )) = dim(X)
(the ‘nullity’ is the dimension of Ker(T ) and the ‘rank’ is the dimension of
Range(T )). By Lemma 1.21, T is injective when dim(Ker(T )) = 0, which
then implies that dim(Range(T )) = dim(X) so that T is onto; similarly, if T
is onto, then dim(Range(T )) = dim(X) which implies that dim(Ker(T )) = 0,
and so T is also injective.
A map is bijective or a bijection if it is both injective and surjective. When
T is a bijection we can define its inverse.
Definition 1.23 A map T : X → Y has an inverse T −1 : Y → X if T is a
bijection, and in this case for each y ∈ Y we define T−1y to be the unique
x ∈ X such that T x = y.
Note that if Ker(T ) = {0}, then the linear map T : X → Range(T ) always
has an inverse; if X is infinite-dimensional T may not map X onto Y, but it
always maps X onto Range(T ), by definition.
The following lemma shows that when T ∈ L(X, Y) has an inverse, the map
T −1 : Y → X is also linear.
Lemma 1.24 A linear map T ∈ L(X, Y) has an inverse if and only if there
exists S ∈ L(Y, X) such that
ST = IX and T S = IY , (1.5)
and then T −1 = S.

1.6 Existence of Bases and Zorn’s Lemma 13
Proof Suppose that T : X → Y is a bijection, so that it has an inverse
T −1 : Y → X; from the definition it follows that T T −1 = IY and T −1T = IX .
It remains to check that T −1 : Y → X is linear; this follows from the injectivity
of T , since
T [T −1
(αy + βz)] = αy + βz = T [αT −1
y + βT −1
z]
therefore implies that
T −1
(αy + βz) = αT −1
y + βT −1
z.
For the converse, we note that T S = IY implies that T : X → Y is onto,
since T (Sy) = y, and that ST = IX implies that T : X → Y is one-to-one,
since
T x = T y ⇒ S(T x) = S(T y) ⇒ x = y.
It follows that if (1.5) holds, then T has an inverse T −1, and applying T −1 to
both sides of T S = IY shows that S = T −1.
Note that if T ∈ L(X, Y) and S ∈ L(Y, Z) are both invertible, then so is
ST ∈ L(X, Z), with
(ST )−1
= T −1
S−1
; (1.6)
since ST is a bijection it has an inverse (ST )−1 such that ST (ST )−1 = IZ ;
multiplying first by S−1 and then by T −1 yields (1.6).
1.6 Existence of Bases and Zorn’s Lemma
We end this chapter by showing that every vector space has a Hamel basis.
To prove this, we will use Zorn’s Lemma, which is a very powerful result that
will allow us to prove various existence results throughout this book. To state
this ‘lemma’ (which is in fact equivalent to the Axiom of Choice, as shown in
Appendix A) we need to introduce some auxiliary concepts.
Definition 1.25 A partial order on a set P is a binary relation on P such
that for a, b, c ∈ P
(i) a a;
(ii) a b and b a implies that a = b; and
(iii) a b and b c implies that a c.

The order is ‘partial’ because two arbitrary elements of P need not be
ordered: consider for example, the case when P consists of all subsets of R
and X Y if X ⊆ Y; one cannot order [0, 1] and [1, 2].
Definition 1.26 Two elements a, b ∈ P are comparable if a b or b a (or
both if a = b). A subset C of P is called a chain if any pair of elements of C
are comparable.
An element b ∈ P is an upper bound for a subset S of P if s b for all
s ∈ S. An element m of P is maximal if m a for some a ∈ P implies that
a = m.
Note that among any finite collection of elements in a chain C there is always
a maximal and a minimal element: if c1, . . . , cn ∈ C, then there are indices
j, k ∈ {1, . . . , n} such that
cj ci ck i = 1, . . . , n; (1.7)
this can easily be proved by induction on n; see Exercise 1.7.
Theorem 1.27 (Zorn’s Lemma) If P is a non-empty partially ordered set in
which every chain has an upper bound, then P has at least one maximal
element.
It is easy to find examples in which there is more than one maximal element.
For example, let P consist of all points in the two disjoint intervals I1 = [0, 1]
and I2 = [2, 3], and say that a b if a and b are contained in the same
interval and a ≤ b. Then every chain in P has an upper bound, and P contains
two maximal elements, 1 and 3.
Theorem 1.28 Every vector space has a Hamel basis.
Proof If V is finite-dimensional, then V has a finite-dimensional basis, by
definition.
So we assume that V is infinite-dimensional. Let P be the collection of all
linearly independent subsets of V . We define a partial order on P by declaring
that E1 E2 if E1 ⊆ E2. If C is a chain in P, then set
E∗
=
E∈C
E.
Note that E∗ is linearly independent, since by (1.7) any finite collection of ele-
ments of E∗ must be contained in one E ∈ C (which is linearly independent).
Clearly E E∗ for all E ∈ C, so E∗ is an upper bound for C.

Exercises 15
It follows from Zorn’s Lemma that P has a maximal element, i.e. a maximal
linearly independent set, and by Lemma 1.13 this is a Hamel basis for V .
As an example of a Hamel basis for an infinite-dimensional vector space, it
is easy to see that the countable set {e( j)}∞
j=1 (as defined in (1.3)) is a Hamel
basis for the space c00 from Example 1.8. However, this is a somewhat artificial
example. We will see later (Exercises 5.7 and 22.1) that no Banach space (the
particular class of vector spaces that will be our main subject in most of the
rest of this book) can have a countable Hamel basis.
Exercises
1.1 Show that if p ≥ 1 and a, b ≥ 0, then
(a + b)p
≤ 2p−1
(ap
+ bp
).
[Hint: find the maximum of the function f (x) = (1 + x)p/(1 + x p).]
1.2 For 1 ≤ p ∞, show that the set L̃ p(0, 1) of all continuous real-valued
functions on (0, 1) for which
ˆ 1
0
| f (x)|p
dx ∞
is a vector space (with the obvious pointwise definitions of addition and
scalar multiplication).
1.3 Show that if E is a basis for a vector space V, then every non-zero v ∈ V
can be written uniquely in the form v = n
j=1 αj ej , for some n ∈ N,
ej ∈ E, and non-zero coefficients αj ∈ K.
1.4 Show that if V has a basis consisting of n elements, then every basis for
V has n elements.
1.5 If T ∈ L(X, Y) show that Ker(T ) and Im(T ) are both vector spaces.
1.6 If X is a vector space over K and U is a subspace of X define an
equivalence relation on X by
x ∼ y ⇔ x − y ∈ U.
The quotient space X/U is the set of all equivalence classes
[x] = x + U := {x + u : u ∈ U}
for x ∈ X. Show that this is a vector space over K if we define
[x] + [y] := [x + y] λ[x] := [λx], x, y ∈ X, λ ∈ K,

and deduce that the quotient map Q : X → X/U given by x → [x] is
linear.
1.7 Show that among any finite collection of elements in a chain C there is
always a maximal and a minimal element: if c1, . . . , cn ∈ C, then there
exist j, k ∈ {1, . . . , n} such that
cj ci ck i = 1, . . . , n.
(Use induction on n.)
1.8 Let Z be a linearly independent subset of a vector space V . Use Zorn’s
Lemma to show that V has a Hamel basis that contains Z.

2
Metric Spaces
Most of the results in this book concern normed spaces; but these are partic-
ular examples of metric spaces, and there are some ‘standard results’ that are
no harder to prove in the more general context of metric spaces. In this chap-
ter we therefore recall the definition of a metric space, along with definitions
of convergence, continuity, separability, and compactness. The treatment in
this chapter is intentionally brisk, but proofs are included. For a more didactic
treatment see Sutherland (1975), for example.
2.1 Metric Spaces
A metric on a set X is a generalisation of the ‘distance between two points’
familiar in Euclidean spaces.
Definition 2.1 A metric d on a set X is a map d : X × X → [0, ∞) that
satisfies
(i) d(x, y) = 0 if and only if x = y;
(ii) d(x, y) = d(y, x) for every x, y ∈ X; and
(iii) d(x, z) ≤ d(x, y) + d(y, z) for x, y, z ∈ X (‘the triangle inequality’).
Even on a familiar space there can be many possible metrics.
Example 2.2 Take X = Kn with any one of the metrics
dp (x, y) =
⎧
⎨
⎩
n
j=1 |x j − yj |p
1/p
1 ≤ p ∞,
maxj=1,...,n |x j − yj | p = ∞.
17

where to go from (2.1) to (2.2) we used the Cauchy–Schwarz inequality
⎛
⎝
n

j=1
ξj ηj
⎞
⎠
2
≤
⎛
⎝
n

j=1
ξ2
j
⎞
⎠
⎛
⎝
n

j=1
η2
j
⎞
⎠ ;
see Exercise 2.1 (and Lemma 8.5 in a more general context).
Note that the space X in the definition of a metric need not be a vector
space. The following example provides a metric on any set X; it is very useful
for counterexamples.
Example 2.3 The discrete metric on any set X is defined by setting
d(x, y) =
0 x = y,
1 x = y.
If A is a subset of X and d is a metric on X, then (A, d|A×A) is another
metric space, where by d|A×A we denote the restriction of d to A × A, i.e.
d|A×A(a, b) = d(a, b) a, b ∈ A; (2.3)
we usually drop the |A×A since this is almost always clear from the context.
If we have two metric spaces (X1, d1) and (X2, d2), then we can choose
many possible metrics on the product space X1 × X2. The most useful choices
are
1

(x1, x2), (y1, y2)

:= d1(x1, y1) + d2(x2, y2) (2.4)
and
2

(x1, x2), (y1, y2)

:=

d1(x1, y1)2
+ d2(x2, y2)2
1/2
. (2.5)
These have obvious generalisations to the product of any finite number of met-
ric spaces. While the expression in (2.4) is simpler and easier to work with, the
definition in (2.5) ensures that the metric on Kn that comes from viewing it as
the n-fold product K × K × · · · × K agrees with the usual Euclidean distance.
Exercise 2.2 provides a larger family p of product metrics.
2.2 Open and Closed Sets
The notion of an open set is fundamental in the study of metric spaces, and
forms the basis of the theory of topological spaces (see Appendix C). We begin
with the definition of an open ball.

20 Metric Spaces
Definition 2.4 If r 0 and a ∈ X we define the open ball of radius r centred
at a as
BX (a,r) := {x ∈ X : d(x, a) r}.
If the space X is clear from the context (as in some of the following
definitions), then we will omit the X subscript.
Definition 2.5 A subset A of a metric space (X, d) is open if for every x ∈ A
there exists r 0 such that B(x,r) ⊆ A. A subset A of (X, d) is closed if
X A is open.
Note that the whole space X and the empty set ∅ are always open, so at the
same time X and ∅ are also always closed. The open ball B(x,r) is open for
any x ∈ X and any r 0 (see Exercise 2.6) and any open subset of X can be
written as the union of open balls (see Exercise 2.7).
Note that in any set X with the discrete metric, any subset A of X is open
(since if x ∈ A, then B(x, 1/2) = {x} ⊆ A) and any subset is closed (since
X A is open).
Lemma 2.6 Any finite intersection of open sets is open, and any union of open
sets is open. Any finite union of closed sets is closed, and any intersection of
closed sets is closed.
Proof We prove the result for open sets; for the corresponding results for
closed sets (which follow by taking complements) see Exercise 2.5.
Let U = ∪α∈AUα, where A is any index set; if x ∈ U, then x ∈ Uα for
some α ∈ A, and then there exists r 0 such that B(x,r) ⊆ Uα ⊆ U, so U is
open.
If U = ∩n
j=1Uj and x ∈ U, then for each j we have x ∈ Uj , and so
B(x,rj ) ⊆ Uj for some rj 0. Taking r = minj rj it follows that
B(x,r) ⊆ ∩n
j=1Uj = U.
In many arguments in this book it will be useful to have a less ‘topological’
definition of a closed set, based on the limits of sequences. We first define what
it means for a sequence to converge in a metric space.
Throughout this book we will use the notation (xn)∞
n=1 for a sequence (to
distinguish it from the set {xn}∞
n=1 in which the order of the elements is irrel-
evant); we will often abbreviate this to (xn), including the index if this is
required to prevent ambiguity, e.g. for a subsequence (xnk )k. We will also fre-
quently abbreviate ‘a sequence (xn)∞
n=1 such that xn ∈ A for every n ∈ N’ to
‘a sequence (xn) ∈ A’.

Definition 2.7 A sequence (xn)∞
n=1 in a metric space (X, d) converges in
(X, d) to x ∈ X if d(xn, x) → 0 as n → ∞. We write xn → x in (X, d)
(or often simply ‘in X’).
For sequences in K we often use the fact that any convergent sequence is
bounded, and the same is true in a metric space, given the following definition.
Definition 2.8 A subset Y of a metric space (X, d) is bounded if there exists1
a ∈ X and r 0 such that Y ⊆ B(a,r), i.e. d(y, a) r for every y ∈ Y.
Any convergent sequence is bounded, since if xn → x, then there exists
N ∈ N such that d(xn, x) 1 for all n ≥ N and so
d(xn, x) ≤ max

1, max
j=1,...,N−1
d(x j , x) for every n ∈ N.
We now describe convergence in terms of open sets.
Lemma 2.9 A sequence (xn) ∈ (X, d) converges to x if and only if for any
open set U that contains x there exists an N such that xn ∈ U for every n ≥ N.
Proof Given any open set U that contains x there exists ε 0 such that
B(x, ε) ⊆ U, and so there exists N such that xn ∈ B(x, ε) ⊆ U for all n ≥ N.
For the other implication, just use the fact that for any ε 0 the set B(x, ε) is
open and contains x.
We can now characterise closed sets in terms of the limits of sequences.
Lemma 2.10 A subset A of (X, d) is closed if and only if whenever (xn) ∈ A
with xn → x it follows that x ∈ A.
Proof Suppose that A is closed and that (xn) ∈ A with xn → x, but x /
∈ A.
Then X A is open and contains x, and so there exists N such that xn ∈ X A
for all n ≥ N, a contradiction.
Now suppose that whenever (xn) ∈ A with xn → x we have x ∈ A, but A
is not closed. Then X A is not open: there exists y ∈ X A and a sequence
rn → 0 such that B(y,rn) ∩ A = ∅. So there exist points yn ∈ B(y,rn) ∩ A,
i.e. a sequence (yn) ∈ A such that yn → y. But then, by assumption, y ∈ A, a
contradiction once more.
1 We could require that a ∈ Y in this definition, since if Y ⊆ B(a,r) with a ∈ X, then for any
choice of a ∈ Y we have d(y, a) ≤ d(y, a) + d(a, a) r + d(a, a).

22 Metric Spaces
2.3 Continuity and Sequential Continuity
We now define what it means for a map f : X → Y to be continuous when
(X, dX ) and (Y, dY ) are two metric spaces. We begin with the ε–δ definition.
Definition 2.11 A function f : (X, dX ) → (Y, dY ) is continuous at x ∈ X if
for every ε 0 there exists δ 0 such that
dX (x
, x) δ ⇒ dY ( f (x
), f (x)) ε.
We say that f is continuous (on X) if f is continuous at every x ∈ X.
Note that strictly there is a distinction to be made between f as a map from
a set X into a set Y, and the continuity of f , which depends on the metrics dX
and dY on X and Y; this distinction is often blurred in practice.
As with simple real-valued functions, continuity and sequential continuity
are equivalent in metric spaces (for a proof see Exercise 2.8).
Lemma 2.12 A function f : (X, dX ) → (Y, dY ) is continuous at x if and only
if f (xn) → f (x) in Y whenever (xn) ∈ X with xn → x in X.
Continuity can also be characterised in terms of open sets, by requiring the
preimage of an open set to be open. This allows the notion of continuity to be
generalised to topological spaces (see Appendix C).
Lemma 2.13 A function f : (X, dX ) → (Y, dY ) is continuous on X if and
only if whenever U is an open set in (Y, dY ), f −1(U) is an open set in (X, dX ),
where
f −1
(U) := {x ∈ X : f (x) ∈ U}
is the preimage of U under f . The same is true if we replace open sets by
closed sets.
Proof Suppose that f is continuous (in the sense of Definition 2.11). Take an
open subset U of Y, and z ∈ f −1(U). Since f (z) ∈ U and U is open in Y,
there exists an ε 0 such that BY ( f (z), ε) ⊆ U. Since f is continuous, there
exists a δ 0 such that x ∈ BX (z, δ) implies that f (x) ∈ BY ( f (z), ε) ⊆ U.
So BX (z, δ) ⊆ f −1(U), i.e. f −1(U) is open.
For the opposite implication, take x ∈ X and set U := BY ( f (x), ε), which
is an open set in Y. It follows that f −1(U) is open in X, so in particular,
BX (x, δ) ⊆ f −1(U) for some δ 0. So

2.4 Interior, Closure, Density, and Separability 23
f (BX (x, δ)) ⊆ BY ( f (x), ε),
which implies that f is continuous.
The result for closed sets follows from the identity
f −1
(Y A) = X f −1
(A) for all A ⊆ Y.
One has to be a little careful with preimages. If f : X → Y, then for U ⊆ X
and V ⊆ Y we have
f −1
( f (U)) ⊇ U and f ( f −1
(V )) ⊆ V.
However, both these inclusions can be strict, as the simple example f : R → R
with f (x) = 0 for every x ∈ R shows: here we have
f −1
( f ([−1, 1])) = f −1
(0) = R and f ( f −1
([−1, 1])) = f (R) = 0.
Also be aware that in general the image of an open set under a continuous
map need not be open, e.g. the image of (−4, 4) under the map x → sin x is
[−1, 1].
2.4 Interior, Closure, Density, and Separability
We recall the definition of the interior A◦ and closure A of a subset of a metric
space. The closure operation allows us to define what it means for a subset
to be dense (A = X) and this in turn gives rise to the notion of separability
(existence of a countable dense subset).
Definition 2.14 If A ⊆ (X, d), then the interior of A, written A◦, is the union
of all open subsets of A.
Note that A◦ is open (since it is the union of open sets; see Lemma 2.6) and
that A◦ = A if and only if A is open.
Lemma 2.15 A point x ∈ X is contained in A◦ if and only if
B(x, ε) ⊆ A for some ε 0.
Proof If x ∈ A◦, then it is an element of some open set U ⊆ A, and then
B(x, ε) ⊆ U ⊆ A for some ε 0. Conversely, if B(x, ε) ⊆ A, then we have
B(x, ε) ⊆ A◦, and so x ∈ A◦.
We will make significantly more use of the closure in what follows.

24 Metric Spaces
Definition 2.16 If A ⊆ (X, d), then the closure of A in X, written A, is the
intersection of all closed subsets of X that contain A.
Note that A is closed (since it is the intersection of closed sets; see Lemma
2.6 again). Furthermore, A is closed if and only if A = A and hence A = A.
Lemma 2.17 A point x ∈ X is contained in A if and only if
B(x, ε) ∩ A = ∅ for every ε 0. (2.6)
It follows that x ∈ A if and only if there exists a sequence (xn) ∈ A such that
xn → x.
Proof We prove the reverse, that x /
∈ A if and only if B(x, ε) ∩ A = ∅ for
every ε 0.
If x /
∈ A, then there is some closed set K that contains A such that x /
∈ K.
Since K is closed, X K is open, and so B(x, ε) ∩ K = ∅ for some ε 0,
which shows that B(x, ε) ∩ A = ∅ (since K ⊇ A).
Conversely, if there exists ε 0 such that B(x, ε) ∩ A = ∅, then x is not
contained in the closed set X B(x, ε), which contains A; so x /
∈ A. This
proves the ‘if and only if’ statement in the lemma.
To prove the final part, if x ∈ A, then (2.6) implies that for any n ∈ N we
have B(x, 1/n)∩ A = ∅, so we can find xn ∈ A such that d(xn, x) 1/n and
thus xn → x. Conversely, if (xn) ∈ A with xn → x, then d(xn, x) ε for n
sufficiently large, which gives (2.6).
Note that in a general metric space
BX (a,r) = {x ∈ X : d(x, a) ≤ r}. (2.7)
If we use the discrete metric from Example 2.3, then BX (a, 1) = {a} for
any a ∈ X, and since {a} is closed we have BX (a, 1) = {a}. However,
{y ∈ X : d(x, a) ≤ 1} = X.
Given the definition of the closure of the set, we can now define what it
means for a set A ⊂ X to be dense in (X, d).
Definition 2.18 A subset A of a metric space (X, d) is dense in X if A = X.
Using Lemma 2.17 an equivalent definition is that A is dense in X if for
every x ∈ X and every ε 0
B(x, ε) ∩ A = ∅,
i.e. there exists a ∈ A such that d(a, x) ε. Another similar reformulation is
that A ∩ U = ∅ for every open subset U of X.

2.5 Compactness 25
Definition 2.19 A metric space (X, d) is separable if it contains a countable
dense subset.
Separability means that elements of X can be approximated arbitrarily
closely by some countable collection {xn}∞
n=1: given any x ∈ X and ε 0,
there exists j ∈ N such that d(x j , x) ε.
For some familiar examples, R is separable, since Q is a countable dense
subset; C is separable since the set ‘Q + iQ’ of all complex numbers of the
form q1 + iq2 with q1, q2 ∈ Q is countable and dense. Since separability of
(X, dX ) and (Y, dY ) implies separability of X ×Y (with an appropriate metric;
see Exercise 2.9), it follows that Rn and Cn are separable.
Separability is inherited by subsets (using the same metric, as in (2.3)). This
is not trivial, since the original countable dense set could be entirely disjoint
from the chosen subset (e.g. Q2 is dense in R2, but disjoint from the subset
{π} × R).
Lemma 2.20 If (X, d) is separable and Y ⊆ X, then (Y, d) is also separable.
Proof We construct A, a countable dense subset of Y, as follows.
Suppose that {xn}∞
n=1 is dense in X; then for each n, k ∈ N, if
B(xn, 1/k) ∩ Y = ∅
then we choose one point from B(xn, 1/k) ∩ Y and add it to A. Constructed in
this way A is (at most) a countable set since we can have added at most N × N
points.
To show that A is dense, take z ∈ Y and ε 0. Now choose k such that
1/k ε/2 and xn ∈ X with d(xn, z) 1/k. Since z ∈ B(xn, 1/k) ∩ Y, we
have B(xn, 1/k) ∩ Y = ∅; because of this there must exist y ∈ A such that
d(xn, y) 1/k and hence
d(y, z) ≤ d(y, xn) + d(xn, z) 2/k ε.
2.5 Compactness
Compactness is an extremely useful property that is the key to many of the
proofs that follow. The most familiar ‘compactness’ result is the Bolzano–
Weierstrass Theorem: any bounded set of real numbers has a convergent
subsequence.
The fundamental definition of compactness in terms of open sets makes
the definition applicable in any topological space (see Appendix C). To state

26 Metric Spaces
this definition we require the following terminology: a cover of a set K
is any collection of sets whose union contains K; given a cover, a sub-
cover is a subcollection of sets from the original cover whose union still
contains K.
Definition 2.21 A subset K of a metric space (X, d) is compact if any cover
of K by open sets has a finite subcover, i.e. if {Oα}α∈A is a collection of open
subsets of X such that
K ⊆
α∈A
Oα,
then there is a finite set {αj }n
j=1 ⊂ A such that
K ⊆
n
j=1
Oαj .
In a metric space compactness in this sense is equivalent to ‘sequential
compactness’, and it is in this form that we will most often make use of com-
pactness in what follows. The equivalence of these two definitions in a metric
space is not trivial; a proof is given in Appendix C (see Theorem C.14).
Definition 2.22 If K is a subset of (X, d), then K is sequentially compact
if any sequence in K has a subsequence that converges and whose limit lies
in K.
(Recall that a subsequence of (xn)∞
n=1 is a sequence of the form (xnk )∞
k=1 where
nk ∈ N with nk+1 nk.)
Using the Bolzano–Weierstrass Theorem we can easily prove the following
basic compactness result.
Theorem 2.23 Any closed bounded subset of K is compact.
Proof First we prove the result for K = R. Take any closed bounded sub-
set A of R, and let (xn) be a sequence in A. Since (xn) ∈ A, we know that
(xn) is bounded, and so it has a convergent subsequence xn j → x for some
x ∈ R. Since xn j ∈ A and A is closed, it follows that x ∈ A and so A is
compact.
Now let A be a closed bounded subset of C and (zn) a sequence in A. If
we write zn = xn + iyn, then, since |zn|2 = |xn|2 + |yn|2, (xn) and (yn) are
both bounded sequences in R. First take a subsequence (zn j )j such that xn j
converges to some x ∈ R. Then take a subsequence of (zn j )j , (zn
j
)j such

2.5 Compactness 27
that yn
j
converges to some y ∈ R; we still have xn
j
→ x. It follows that
zn
j
→ x + iy, and since zn
j
∈ A and A is closed it follows that x + iy ∈ A,
which shows that A is compact.
Compact subsets of metric spaces are closed and bounded.
Lemma 2.24 If K is a compact subset of a metric space (X, d), then K is
closed and bounded.
Proof If (xn) ∈ K and xn → x, then any subsequence of (xn) also converges
to x. Since K is compact, it has a subsequence xn j → x with x ∈ K. By
uniqueness of limits it follows that x = x and so x ∈ K, which shows that K
is closed (see Lemma 2.10).
If K is compact, then the cover of K by the open balls {B(k, 1) : k ∈ K}
has a finite subcover by balls centred at {k1, . . . , kn}. Then for any k ∈ K we
have k ∈ B(kj , 1) for some j = 1, . . . , n and so
d(k, k1) ≤ d(k, kj ) + d(kj , k1) 1 + max
j=2,...,n
d(x j , x1).
Lemma 2.25 If (X, d) is a compact metric space, then a subset K of X is
compact if and only if it is closed.
Proof If K is a compact subset of (X, d), then it is closed by Lemma 2.24. If
K is a closed subset of a compact metric space, then any sequence in K has a
convergent subsequence; its limit must lie in K since K is closed, and thus K
is compact.
We will soon prove in Theorem 2.27 that being closed and bounded
characterises compact subsets of Rn, based on the following observation.
Theorem 2.26 If K1 is a compact subset of (X1, d1) and K2 is a compact
subset of (X2, d2), then K1 × K2 is a compact subset of the product space
(X1 × X2, p), where p is any of the product metrics from Exercise 2.2.
Proof Suppose that (xn, yn) ∈ K1 × K2. Then, since K1 is compact, there is
a subsequence (xn j , yn j ) such that xn j → x for some x ∈ K1. Now, using the
fact that K2 is compact, take a further subsequence, (xn
j
, yn
j
) such that we
also have yn
j
→ y for some y ∈ K2; because xn
j
is a subsequence of xn j we
still have xn
j
→ x. Since
p((xn
j
, yn
j
), (x, y)) =

d1(xn
j
, x)p
+ d2(yn
j
, y)p
1/p
it follows that (xn
j
, yn
j
) → (x, y) ∈ K1 × K2.

28 Metric Spaces
By induction this shows that the product of any finite number of compact
sets is compact. (Tychonoff’s Theorem, proved in Appendix C, shows that the
product of any collection of compact sets is compact when considered with an
appropriate topology.)
Theorem 2.27 A subset of Kn (with the usual metric) is compact if and only
if it is closed and bounded.
Note that Kn with the usual metric is given by the product K × · · · × K,
using 2 to construct the metric on the product.
Proof That any compact subset of Kn is closed and bounded follows immedi-
ately from Lemma 2.24.
For the converse, note that it follows from Theorem 2.26 that
Qn
M := {x ∈ Kn
: |x j | ≤ M, j = 1, . . . , n}
is a compact subset of Kn for any M 0. If K is a bounded subset of Kn, then
it is a subset of Qn
M for some M 0. If it is also closed, then it is a closed
subset of a compact set, and hence compact (Lemma 2.25).
We now give three fundamental results about continuous functions on
compact sets.
Theorem 2.28 Suppose that K is a compact subset of (X, dX ) and that
f : (X, dX ) → (Y, dY ) is continuous. Then f (K) is a compact subset of
(Y, dY ).
Proof Let (yn) ∈ f (K). Then yn = f (xn) for some xn ∈ K. Since (xn) ∈ K
and K is compact, there is a subsequence of xn that converges to some x∗ ∈ K,
i.e. xn j → x∗ ∈ K. Since f is continuous it follows (using Lemma 2.12) that
as j → ∞
yn j = f (xn j ) → f (x∗
) =: y∗
∈ f (K),
i.e. the subsequence yn j converges to some y∗ ∈ f (K). It follows that f (K)
is compact.
The following is an (almost) immediate corollary.
Proposition 2.29 Let K be a compact subset of (X, d). Then any continuous
function f : K → R is bounded and attains its bounds, i.e. there exists an
M 0 such that | f (x)| ≤ M for all x ∈ K, and there exist x, x ∈ K such
that

2.5 Compactness 29
f (x) = inf
x∈K
f (x) and f (x) = sup
x∈K
f (x). (2.8)
Proof Since f is continuous and K is compact, f (K) is a compact subset
of R, so f (K) is closed and bounded (by Theorem 2.27); in particular, there
exists M 0 such that | f (x)| ≤ M for every x ∈ K. Since f (K) is closed, it
follows that
sup {y : y ∈ f (K)}
(see Exercise 2.12) and so there exists x as in (2.8). The argument for x is
almost identical.
Finally, any continuous function on a compact set is also uniformly contin-
uous.
Lemma 2.30 If f : (X, dX ) → (Y, dY ) is continuous and X is compact, then
f is uniformly continuous on X: given ε 0 there exists δ 0 such that
dX (x, y) δ ⇒ dY ( f (x), f (y)) ε x, y ∈ X.
Proof If f is not uniformly continuous, then there exists ε 0 such that for
every δ 0 we can find x, y ∈ X with dX (x, y) ε and dY ( f (x), f (y)) ≥ ε.
Choosing xn, yn for each δ = 1/n we obtain xn, yn such that
dX (xn, yn) 1/n and dY ( f (xn), f (yn)) ≥ ε. (2.9)
Since X is compact, we can find a subsequence xn j such that xn j → x with
x ∈ X. Since
dX (yn j , x) ≤ dX (yn j , xn j ) + dX (xn j , x),
it follows that yn j → x also. Since f is continuous at x, we can find δ 0
such that dX (z, x) δ ensures that dY ( f (z), f (x)) ε/2. But then for j
sufficiently large we have dX (xn, j , x) δ and dX (yn j , x) δ, which implies
that
dY ( f (xn j ), f (yn j )) ≤ dY ( f (xn j ), f (x)) + dY ( f (x), f (yn j )) ε,
contradicting (2.9).
We often apply this when f : (K, dX ) → (Y, dY ) and K is a compact subset
of a larger metric space (X, dX ).

30 Metric Spaces
Exercises
2.1 Using the fact that n
j=1(ξj − ληj )2 ≥ 0 for any ξ, η ∈ Rn and any
λ ∈ R, prove the Cauchy–Schwarz inequality
⎛
⎝
n

j=1
ξj ηj
⎞
⎠
2
≤
⎛
⎝
n

j=1
ξ2
j
⎞
⎠
⎛
⎝
n

j=1
η2
j
⎞
⎠ .
(For the usual dot product in Rn this shows that |x · y| ≤ xy.)
2.2 Suppose that (X j , dj ), j = 1, . . . , n are metric spaces. Show that
X1 × · · · × Xn is a metric space when equipped with any of the product
metrics p defined by setting
p((x1, . . . , xn),(y1, . . . , yn))
:=
⎧
⎨
⎩
!
n
j=1 dj (x j , yj )p
1/p
, 1 ≤ p ∞
maxj=1,...,n dj (x j , yj ), p = ∞.
(You should use the inequality
#
(α1 + β1)p
+ (α2 + β2)p
$1/p
≤ (α
p
1 +α
p
2 )1/p
+(β
p
1 +β
p
2 )1/p
, (2.10)
which holds for all α1, α2, β1, β2 ∈ R and 1 ≤ p ∞; we will prove
this in Lemma 3.6 in the next chapter.)
2.3 Show that if d is a metric on X, then so is
d̂(x, y) :=
d(x, y)
1 + d(x, y)
.
(This new metric makes (X, d̂) into a bounded metric space.) [Hint: the
map t → t/(1 + t) is monotonically increasing in t.]
2.4 Let s(K) be the space of all sequences x = (x j )∞
j=1 with x j ∈ K
(bounded or unbounded). Show that
d(x, y) :=
∞

j=1
2− j |x j − yj |
1 + |x j − yj |
defines a metric on s. If (x(n))n is any sequence in s show that x(n) → y
in this metric if and only if x
(n)
j → yj for each j ∈ N. (Kreyszig, 1978)
2.5 Show that any finite union of closed sets is closed and any intersection
of closed sets is closed.
2.6 Show that if x ∈ X and r 0, then B(x,r) is open.

Exercises 31
2.7 Show that any open subset of a metric space (X, d) can be written as the
union of open balls. [Hint: if U is open, then for each x ∈ U there exists
r(x) 0 such that x ∈ B(x,r(x)) ⊆ U.]
2.8 Show that a function f : (X, dX ) → (Y, dY ) is continuous if and only if
f (xn) → f (x) whenever (xn) ∈ X with xn → x in (X, d).
2.9 Show that if (X, dX ) and (Y, dY ) are separable, then (X × Y, p) is
separable, where p is any one of the metrics from Exercise 2.2.
2.10 Suppose that {Fα}α∈A are a family of closed subsets of a compact metric
space (X, d) with the property that the intersection of any finite number
of the sets has non-empty intersection. Show that ∩α∈A Fα is non-empty.
2.11 Suppose that (Fj ) is a decreasing sequence [Fj+1 ⊆ Fj ] of non-empty
closed subsets of a compact metric space (X, d). Use the result of the
previous exercise to show that ∩∞
j=1 Fj = ∅.
2.12 Show that if S is a closed subset of R, then sup(S) ∈ S.
2.13 Show that if f : (X, dX ) → (Y, dY ) is a continuous bijection and X is
compact, then f −1 is also continuous (i.e. f is a homeomorphism).
2.14 Any compact metric space (X, d) is separable. Prove the stronger result
that in any compact metric space there exists a countable subset (x j )∞
j=1
with the following property: for any ε 0 there is an M(ε) such that for
every x ∈ X we have
d(x j , x) ε for some 1 ≤ j ≤ M(ε).

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Title: Selling Things
Author: Orison Swett Marden
Joseph Francis MacGrail
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Language: English
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*** START OF THE PROJECT GUTENBERG EBOOK SELLING THINGS
***

Orison S. Marden
SELLING THINGS

BY
ORISON SWETT MARDEN
AUTHOR OF “PUSHING TO THE FRONT,” “PEACE, POWER AND
PLENTY,” “THE VICTORIOUS ATTITUDE,” ETC.
WITH THE ASSISTANCE OF
JOSEPH F. MacGRAIL
INSTRUCTOR IN SALESMANSHIP AND EFFICIENCY FOR MANY
LARGE SALES AND INDUSTRIAL ORGANIZATIONS
NEW YORK
THOMAS Y. CROWELL COMPANY
PUBLISHERS
Copyright, 1916,
By THOMAS Y. CROWELL COMPANY
Thirteenth Thousand

TO MY FRIEND
CHARLES M. SCHWAB
THE MASTER EXECUTIVE, PRODUCER, SALESMAN.

CONTENTS
CHAPTER PAGE
I The Man Who Can Sell Things 1
II Training the Salesman 6
III The Most Important Subjects of Study 14
IV Making a Favorable Impression 19
V The Selling Talk or “Presentation” 28
VI The Approach and Expression 33
VII The Ability to Talk Well 37
VIII How to Get Attention 42
IX Tact as a Friend-Winner and Business-Getter 47
X Sizing Up the Prospect 62
XI How Suggestion Helps in Selling 71
XII The Force of Cheerful Expectancy 79
XIII The Gentle Art of Persuasion 86
XIV Helping the Customer to Buy 94
XV Closing the Deal 105
XVI The Greatest Salesman—Enthusiasm 112
XVII The Man at the Other End of the Bargain 119
XVIII Meeting and Forestalling Objections 125
XIX Quality as a Salesman 133
XX A Salesman’s Clothes 139
XXI Finding Customers 148
XXII When You Are Discouraged 155
XXIII The Stimulus of Rebuffs 163
XXIV Meeting Competition: “Know Your Goods” 177
XXV The Salesman and the Sales Manager 184
XXVI Are You a Good Mixer? 189
XXVII Character Is Capital 207

XXVIII The Price of Mastership 213
XXIX Keeping Fit and Salesmanship 226
Appendix—Sales Pointers 250
SELLING THINGS

CHAPTER I
THE MAN WHO CAN SELL THINGS
Cultivate all the arts and all the helps to mastership.
The world always listens to a man with a will in him.
Soon after Henry Ward Beecher went to Plymouth Church he
received a letter from a Western parish, asking him to send them a
new pastor. After describing the sort of man they wanted, the letter
closed with the following injunction: “Be sure to send us a man who can
swim. Our last pastor was drowned while fording the river, on a visit
to his parishioners.”
Now, this is the sort of a man that is wanted everywhere, in every
line of human activity, the man who can swim, the salesman who
can swim, who can sell things, who can go out and get business, the
man who can take a message to Garcia, who can bring back the
order, the man who can “deliver the goods.”
The whole business world to-day is hunting for the man who can
sell things; there is a sign up at every manufacturing establishment,
every producing establishment for the man who can market
products. There is nobody in greater demand than the efficient
salesman, and he is rarely if ever out of a job.
Only a short while ago two companies actually went to law about
a salesman who transferred his connection from one to the other, his
original employers holding that he had no right to do so, as he was
under contract (at a $50,000 salary) to them.
In spite of the fact that thousands of employees are looking for
positions, on every hand we see employers looking for somebody
who can “deliver the goods”; a salesman who will not say that if

conditions were right, if everything were favorable, if it were not for
the panic, or some other stumbling block, he could sell the goods.
Everywhere employers are looking for some one who can do things,
no matter what the conditions may be.
There is no place in salesmanship for the man who waits for
orders to come to him. He is simply an order taker, not a salesman.
Live men, men with vigorous initiative and lots of pluck and grit,
men who can go out and get business are wanted.
It should not be necessary to prove that training is needed for
success in salesmanship or in any business. Yet, because men have
been compelled for centuries “to learn by their mistakes,” to pick up
here and there, by hard knocks, a little knowledge about their work,
there has been a prejudice against trying to teach business by sane,
scientific methods. Besides, in former times, the working man and
the mere merchant were supposed to belong to a low class of
society, apart from the noble and the learned, and little attention
was given to their needs. A man, too, was believed to be born with a
natural aptitude for salesmanship or business building, and this was
supposed to be all-sufficient.
To-day there are many men and women attracted by the big
profits in salesmanship, who would like to become salesmen and
saleswomen, but they feel they have not this natural aptitude to
insure permanent success.
It is true that, just as certain men and women are born with
natural gifts for music and for art, so certain men and women have,
in a high degree, the natural qualities which enable them to succeed
in selling either their brain power or merchandise. But while it is true
that some people have more natural capacity than others, it is not
true to-day, and it was never true in the fine arts, in athletics, or in
commercial pursuits, that the untrained man is the equal of the
trained man.
Man is always improving Nature, or, if you prefer, he is always
helping Nature. Central Park, New York, is more beautiful because

the landscape gardener has been helping Nature; the farmer is the
reaper of bigger and better crops because he is following the advice
of the chemist, who tells him how to fertilize the soil; the Delaware
River and Hell Gate have become more easily navigable, because the
engineer has removed obstacles which Nature had placed in those
waters; Colorado’s arid lands are irrigated, thanks to the skill of the
civil engineer; the horticulturist aids Nature by grafting and pruning;
the scientist comes to the help of human nature with antiseptic
methods in surgery; and the inventor shows Nature how electricity
can be put to numberless practical uses.
Let us not fool ourselves; we need to study, we need to be trained
for every business in life. And in these days the training by which
natural defects are overcome and natural aptitude is developed into
effective ability can be obtained by every youth. No matter how
great your natural ability in any direction, in order to get the best
results, it must be reënforced by this special training.
The untrained man may get results here and there because he has
natural ability and unconsciously uses the right methods. The trained
man is getting results regularly because he is consistently using the
right methods.
Business men no longer attribute a lost sale, where it should have
been made, to “hard luck,” but to ignorance of the science of
salesmanship.
The “born” salesman is not as much in vogue as formerly.
Business is becoming a science, and almost any honest, dead-in-
earnest, determined youth can become an expert in it, if he is willing
to pay the price.
It is scientific salesmanship to-day, and not luck, that gets the
order.

CHAPTER II
TRAINING THE SALESMAN
The consciousness of being superbly equipped for your work
brings untold satisfaction.
Efficiency is the watchword of to-day. The half-prepared man,
the man who is ignorant, the man who doesn’t know his lines, is
placed at a tremendous disadvantage.
A student seeking admission to Oberlin College asked its famous
president if there was not some way of taking a sort of homeopathic
college course, some short-cut by which he could get all the
essentials in a few months.
This was the president’s reply: “When the Creator wanted a
squash, he created it in six months, but when he wanted an oak, he
took a hundred years.”
One of the highest-paid women workers in the world, the foreign
buyer for a big department store, owes her position more to
thorough training for her work than to any other thing. Between
salary and commissions, her income amounts to thirty thousand
dollars a year. Speaking of her place in the firm, one of its highest
members said to a writer: “We regard Miss Blank as more of a friend
than an employee; and she came to us just twenty years ago with
her hair in pig-tails, tied with a shoe string; and she was so ill fed
and ill clothed we had to pass her over to our house nurse to get her
currycombed and scrubbed before we could put her on as a cash
girl. Without training, she would probably have dropped back in the
gutter as an unfit and a failure. With training, she has become one
of the ablest business women in the country.”

There are a thousand pigmy salesmen to one Napoleon salesman;
but if you have natural ability for the marketing of any of the great
products of the world, all you need to make you a Napoleon
salesman is sound training and willingness to work faithfully. With
such a foundation for success you will not long be out of a job, or
remain in obscurity, for wherever you go, no matter how hard the
times, you will see an advertisement for just such a man.
The term “salesmanship” is a very broad one; it covers many
fields. The drummer for a boot and shoe house, the insurance agent
and manager, the banker and broker, whose business is to dispose of
millions of dollars’ worth of stocks and bonds—all these are
“salesmen,” trafficking in one kind of goods or another—all form a
part of the world’s great system of organized barter.
There are three essentials which must be considered in deciding
on salesmanship or any other vocation, namely: taste, talent, and
training. The first is, by far, the most important of these essentials,
for whatever we have a taste for, we will be interested in; what we
really become interested in, we are bound to love, sooner or later,
and success comes from loving our work.
To find out whether or not you are cut out for a salesman, you
must first analyze the question of your taste and your talent. In this
matter, however, it should be borne in mind that human nature,
especially in youth, is plastic, and that we can be molded by others,
or we can mold ourselves. Even though one has not a strong taste,
naturally, or a decided talent for salesmanship, he can acquire both,
for even talent, like taste, may be either natural or acquired. By
proper training in salesmanship, which means the right kind of
reading, observing and listening, and right practicing, we can
develop our taste and ability so as to become good salesmen or
good saleswomen.
The basic requirements for successful salesmanship are good
health, a cheerful disposition, courtesy, tact, resourcefulness, facility
of expression, honesty, a firm and unshakable confidence in one’s
self, a thorough knowledge of, and confidence in, the goods which

one is selling, and ability to close. True cordiality of manner must be
reënforced by intelligence and by a ready command of information in
regard to the matters in hand. It will be seen that all these things
make the man as well as the salesman—when coupled with sincerity
and highmindedness, they can’t but bring success in any career.
The foundation for salesmanship can hardly be laid too early. The
youth who uses his spare time when at school, in vacation season,
and out of business hours, in acquiring the art of salesmanship will
gain power to climb up in the world that cannot be obtained so
readily by any other means.
Fortunate is the young man who has received the right kind of
business training. No matter what his occupation or profession, such
training will make him a more efficient worker. Many youths have
had fathers whose experience and advice have been valuable to
them. Others have been favored by getting into firms of high caliber.
As a result they have been in a splendid environment during their
most formative years, and in so far have had an inestimable
advantage in success training.
Many people have the impression that almost anybody can be a
salesman, and that salesmanship doesn’t require much, if any,
special training. The young man who starts out to sell things on this
supposition will soon find out his mistake. If salesmanship is to be
your vocation you cannot afford to take any such superficial view of
its requirements. You cannot afford to botch your life. You cannot
afford a little, picayune career as a salesman, with a little salary and
no outlook. If salesmanship is worth giving your life to, it is worth
very serious and very profound and scientific preparation and
training.
I know a physician, a splendid fellow, who studied medicine in a
small, country medical school, where there was very little material,
and practically no opportunity for hospital work. In fact, during his
years of preparation his experience outside of medical books was
very meager. Since getting his M. D. diploma this man has been a
very hard worker and has managed to get a fair living, but he is

much handicapped in his chance to make a name in his profession.
He has a fine mind, however, and if he had gone to the Harvard
Medical School in Boston, or to one of the other great medical
schools where there is an abundance of material for observation and
facilities for practice in the hospitals and clinics, he would have
learned more in six months, outside of what he gathered from books
and lectures, than he learned in all of his course in the country
medical schools. His poor training has condemned him to a mediocre
success, when his natural ability, with a thorough preparation, would
have made him a noted physician.
You cannot afford to carry on your life work as an amateur, with
improper preparation. You want to be known as an expert, as a man
of standing, a man who would be looked up to as an authority, a
specialist in his line. To enter on your life work indifferently
prepared, half trained, would be like a man going into business
without even a common school education, knowing nothing about
figures. No matter how naturally able such a man might be, people
would take advantage of his ignorance. He would be at the mercy of
his bookkeeper and other employees, and of unscrupulous business
men. And if he should try to make up for his lack of early training or
education, he must do it at a great cost in time and energy.
Successful salesmanship of the highest order requires not only a
fine special training, but also a good education and a keen insight
into human nature; it also requires resourcefulness, inventiveness
and originality. In fact, a salesman who would become a giant in his
line, must combine with the art of salesmanship a number of the
highest intellectual qualities.
Yet in salesmanship, as in every other vocation, there is not one
qualification needed that can not be cultivated by any youth of
average ability and intelligence. Success in it, as in every other
business and profession, is merely the triumph of the common
virtues and ordinary ability.
In salesmanship, as in war, there is offensive and defensive. The
trained salesman knows how to attack, and he knows how to defend

himself when he is attacked. Everything contained within the covers
of this book has for its object the most effective offensive and
defensive methods in selling.

CHAPTER III
THE MOST IMPORTANT SUBJECTS OF STUDY
“Salesmanship is knowing yourself, your company, your
prospect and your product, and applying your knowledge.”
The qualities which make a great business man also enter into
the making of a great salesman.
Salesmanship is fast becoming a profession, and only the
salesman who is superbly equipped can hope to win out in any
large way.
Different authorities agree pretty much on the subjects which
must be studied or understood in the making of good salesmen,
although they classify in somewhat different ways the headings
under which salesmanship should be studied.
Mr. Arthur F. Sheldon, for instance, in his able Course, has divided
the knowledge pertaining to scientific salesmanship under four
heads: 1, The Salesman; 2, The Goods; 3, The Customer; 4, The
Sale. The “Drygoods Economist” has some excellent courses on
salesmanship, in which they use almost this identical classification,
treating the subject under the four general divisions: 1, The
Salesman; 2, The Goods; 3, The Customer; 4, Service. Mr. Charles L.
Huff has added to the valuable data on salesmanship a book in
which he gives the following five factors as the headings under
which the subject of salesmanship should be covered, namely: 1,
Price; 2, Quality; 3, Service; 4, Friendship; 5, Presentation.
Every salesman is really teaching the customer something about
the goods. He is, so to speak, a teacher of values, or if you prefer, “a
business missionary.” In order to teach well he should have these
most valuable assets: first, right methods of meeting customer;

second, thorough knowledge of self, of goods, of customer and
conditions; third, ability to meet competition, both real and
imaginary; fourth, helpful habits; fifth, good powers of originating
and planning; sixth, a selling talk, or something worth while saying;
seventh, properly developed feelings, which will add force to what
he says.
In a brief and helpful course on salesmanship “System,” a business
magazine, gives great emphasis to the value of dwelling on five
buying motives—1, Money; 2, Utility; 3, Caution; 4, Pride; 5, Self-
indulgence, or Yielding to Weakness.
If a salesman will keep before his mind these five points, and if he
appeals to the human traits they indicate he will become a master in
closing deals.
A great many methods are used to-day for rating employees, just
as Dun and Bradstreet rate firms. According to Roger W. Babson,
there is a Mr. Horner, of Minneapolis, who rates his salesmen and
trains them along these lines:
HABITS OF WORK
1. Idealism
2. Intelligence
a. Understanding of business
b. Selecting Policy to suit age
and condition of applicant
c. Self-culture.
3. Hopefulness
4. Optimism
5. Uniform courtesy
a. To clients
b. To office force
c. To fellow agents

6. Number of daily interviews
7. Concentration or effectiveness of
work, as to waste of time or
energy.
8. Loyalty
a. To company
b. To organization
c. To fellow agents
9. Attention to old policy holders
10. Enthusiasm.
A final and very vital point to consider is this: Why do salesmen
meet opposition?
Mr. Huff, in his very practical and interesting book on
salesmanship, has classified under six general heads the causes of
opposition. These are: First, Prior Dissatisfaction; Second, General
Prejudice; Third, Buyer’s Mood; Fourth, Conservatism; Fifth, Bad
Business; Sixth, Personal Dislike for Salesman.
It is up to the salesman to analyze the customer and decide just
which of these six points of opposition is causing him to lose
business.
Just in the degree that he can locate the exact trouble, and then
overcome it in the proper way, will he be able to get the business
which may seem at first absolutely beyond him.
Any or all of these six causes of opposition will not overwhelm the
master salesman, but the mediocre or indifferent salesman is bound
to collapse when confronted with any one of them. And if he does
not train himself to meet and overcome opposition he is doomed to
failure, or at least to a very poor grade of success—not worthy the
name.

Remember, Mr. Salesman, it is always up to you. Develop your
brain power, and then use that power for all it is worth.

CHAPTER IV
MAKING A FAVORABLE IMPRESSION
Go boldly; go serenely, go augustly;
Who can withstand thee then!—Browning.
The personality of a salesman is his greatest asset.
A Washington government official called on me some time ago,
and before he had reached my desk I knew he was a man of
importance, on an important mission. He had that assured bearing
which indicated that he was backed by authority—in this instance
the authority of the United States—and the dignity of his bearing
and manner commanded my instant respect and attention.
The impression you make as you enter a prospect’s office will
greatly influence the manner of your reception. It is imperative to
make a favorable first impression, otherwise you will have to spend
much valuable time and energy and suffer a great deal of
embarrassment in trying to right yourself in your prospect’s
estimation, because he will not do business with you until you have
made a favorable impression on him.
Some salesmen approach their prospect with such an apologetic,
cringing, “excuse me for taking up your valuable time” air, that they
give him the idea they are not on a very important mission, and that
they are not sure of themselves, that they have not much confidence
in the firm they represent or the merchandise they are trying to sell.
Approach the one with whom you expect to do business like a
man, without any doubts, without any earmarks of a cringing,
crawling or craven disposition. Enter his office as the Washington
official entered mine, like a high-class man meeting a high-class
man. You will compel attention and respect instantly, as he did.

Your introduction is an entering wedge, your first chance to score
a point. If you present a pleasing picture as you enter you will score
a strong point. Here is where you must choose the golden mean
between cringing and over-boldness. If you approach a man with
your hat on, and a cigar or cigarette in your mouth, or still smoking
in your fingers; if your breath smells of liquor; if you show that you
are not up to physical standard; if there is any evidence of
dissipation in your appearance; if you swagger or show any lack of
respect, all these things will count against you. If you present an
unpleasing picture, if there is anything about you which your
prospect does not like; if you bluster, or if you lack dignity; if you do
not look him straight in the eye; if there is any evidence of doubt or
fear or lack of confidence in yourself, you will at once arouse a
prejudice in his mind that will cause him to doubt the story you tell
and to look with suspicion at the goods you are trying to sell.
A salesman once entered a business man’s office holding a tooth-
pick in his mouth. You may think it was a little thing, but it so
prejudiced the would-be customer against him at the start that it
made it much more difficult for him even to get a chance to show his
samples. The business man in question was very particular in regard
to little points of manners, and was himself a model of deportment.
I know of another salesman who makes a most unfortunate first
impression because he has no presence whatever, not a particle of
dignity; he is timid and morbidly self-conscious, and it takes him
some minutes after he has met a stranger to regain his self-
possession. To those who know him he is a kindly and genuinely
lovable man, but he does not appear to advantage at a first
introduction. He is a college graduate, and was so popular and stood
so high in his class that he was proposed to represent it at
commencement. He was defeated, however, on the plea that he
would make such a bad impression on the public that he would not
properly represent the class.
Self-possession is an indispensable quality in a salesman. It is
natural to the man who has confidence in himself, and without self-

confidence it is hard to make a dignified appearance or to make
others believe in you.
What you think of yourself will have a great deal to do with what a
prospect will think of you, because you will radiate your estimate of
yourself. If you have a little seven-by-nine model of a man in your
mind you will etch that picture on the mind of your prospect. In
approaching a prospect, walk, talk and act not only like a man who
believes in himself, but one who also believes in and thoroughly
knows his business. When a physician is called into a home in an
emergency, no matter how able a man may be at the head of the
house, no matter how well educated the mother and children may
be, everybody stands aside when he enters. They feel that the
doctor is the master of the situation, that he alone knows what to
do, and they all defer to him. Everybody follows his directions
implicitly.
You should approach a possible customer with something of this
professional air, an air of supreme assurance, of confidence in your
ability, in your honesty and integrity, confidence in your knowledge
of your business. Your professional dignity alone will help to make a
good impression, and will win courtesy. It will insure you at least a
respectful hearing, and there is your chance to play your part in a
masterful manner.
A publisher who has a large number of book agents in the field,
advises his men to act, when the servant answers the door bell, as
though they were expected and welcome. He tells them, if it is
raining to take off their rubbers, if it is muddy or dusty to wipe off
their shoes and act as though they expected to go in.
The idea is to make a favorable impression upon the servant first
of all, for if they were to behave as though they were not sure they
would be admitted, apologizing for making so much trouble and
assuming the attitude of asking a favor, they would communicate
their doubt to the servant, and would not be likely to gain
admittance, not to speak of an audience with the mistress. In short,

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