Graduate texts in mathematics analysis now

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Graduate Texts in Mathematics 118
Editorial Board
J.H. E\\'ing

Springer
New York
Berlin
Heidelberg
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo

FW Gehring

P.R. Halmos


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Graduate Texts in Mathematics

2

3
4
5
6
7
8
9

IO
11

12
13

14
15
16
17
18
19
20
21
22
23
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29

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32

TAKEUTJIZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
OXTOBY. Measure and Category. 2nd ed.
SCHAEFER. Topological Vector Spaces.
HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
MAC LANE. Categories for the Working
Mathematician.
HUGHES!PIPER. Projective Planes.
SERRE. A Course m Arithmetic.
TAKEU'llIZARING. Axiomatic Set Theory.
HUMPHREYS. Introduction to Lie Algebras
and Representation Theory.
CollEN. A Course in Simple Homotopy
Theory.
CONWAY. Functions of One Complex
Variable L 2nd ed.
BEALS. Advanced Mathematical Analysis.
ANDERSON!FuLLER. Rings and Categories
of Modules. 2nd ed.
GoLUBITSKy/GUILLEMIN. Stable Mappings
and Their Smgularities.
BERBERIAN. Lectures in Functional
Analysis and Operator Theory.
WINTER. The Structure of Fields.
ROSENBLATT. Random Processes. 2nd ed.

HALMos. Measure Theory.
HALMos. A Hilbert Space Problem Book.
2nd ed.
HUSEMOLLER. Fibre Bundles. 3rd ed.
HUMPHREYS. LineM Algebraic Groups.
BARNESIMACK. An Algebraic Introduction
to Mathematical Logic.
GREUB. Lmear Algebra. 4th ed.
HOLMES. Geometric Functional Analysis
and Its Applications.
HEWITT/STROMBERG. Real and Abstract
Analysis.
MANEs. Algebraic Theories.
~Y. General Topology.
ZARIsKJISAMUEL. Commutative Algebra.
VoU.
ZARlSKJISAMUEL. Commutative Algebra.
Vol.II.
JACOBSON. Lectures in Abstract Algebra L
Basic Concepts.
JACOBSON. Lectures in Abstract Algebra
IL LineM Algebra.
JACOBSON. Lectures in Abstract Algebra
m. Theory of Fields and Galois Theory.

33 HIRSCH. Differential Topology.
34 SPITZER. Principles of Random Walk.
2nd ed.
35 WERMER. Banach Algebras and Several
Complex Variables. 2nd ed.

36 ~~N~oKAetal.LmeM
Topological Spaces.
37 MONK. Mathematical Logic.
38 GRAUERTIFRmsCHE. Several Complex
Variables.
39 AltvESON. An Invi.tation to C* -Algebras.
40 KEMENY/SNE.U.!KNAPP. Denumerable
Markov Chains. 2nd ed.
41 APoSTOL. ModulM Functions and
Dirichlet Series in Number Theory.
2nd ed.
42 SERRE. LineM Representations of Finite
Groups.
43 GILLMANIJERISON. Rings of Continuous
Functions.
44 KENDIG. Elementary Algebraic Geometry.
45 LoEVE. Probability Theory L 4th ed.
46 LoEVE. Probability Theory II. 4th ed.
47 MOISE. Geometric Topology in
Dimensions 2 and 3.
48 SACHslWu. General Relativity for
Mathematicians.
49 GRUENBERGIWEIR. LmeM Geometry.
2nd ed.
50 EDWARDS. Fennal's Last Theorem.
51 KLiNGENBERG. A Course m Differential
Geometry.
52 HARTSHORNE. Algebraic Geometry.
53 MANlN. A Course m Mathematical Logic.
54 GRAVERIWATIGNS. Combinatorics with

Emphasis on the Theory of Graphs.
55 BROWNlPEARcy. Introduction to Operator
Theory I: Elements of Functional
Analysis.
56 MAsSEY. Algebraic Topology: An
Introduction.
57 CRoWELL/Fox. Introduction to Knot
Theory.
58 KOBLITZ. p-adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 ARNOll. Mathematical Methods in
Oassical Mechanics. 2nd ed.

continued after index


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Gert K. Pedersen

Analysis Now

Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo


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Gert K. Pedersen

K0benhavns Universitets Matematiske Institut
2100 K0benhavn 0
Denmark
Editorial Board

J.H. Ewing
Department of
Mathematics
Indiana University
Bloomington, IN 47401
USA

F.W. Gehring
Department of
Mathematics
University of Michigan
Ann Arbor, MI 48109
USA

P.R. Halmos
Department of
Mathematics
Santa Clara University
Santa Clara, CA 95053
USA

Mathematics Subject Classification (1980): 46-01, 46-C99
Library of Congress Cataloging-in-Publication Data
Pedersen, Gert Kjrergard.
Analysis now / Gert K. Pedersen.

p.
cm.-(Graduate texts in mathematics; 118)
Bibliography: p.
Includes index.
1. Functional analysis. I. Title. II. Series.
QA320.P39 1988
88-22437
515.7-dcI9
Printed on acid-free paper.

© 1989 by Springer-Verlag New York Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia.
Printed in the United States of America.
9 8 7 6 5 4 3 2 I
ISBN 0-387-96788-5 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540-96788-5 Springer-Verlag Berlin Heidelberg New York


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For

DIu! and Cecilie,
innocents at home


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Preface

Mathematical method, as it applies in the natural sciences in particular,
consists of solving a given problem (represented by a number of observed or
observable data) by neglecting so many of the details (these are afterward
termed "irrelevant") that the remaining part fits into an axiomatically estab­
lished model. Each model carries a theory, describing the implicit features of
the model and its relations to other models. The role of the mathematician
(in this oversimplified description of our culture) is to maintain and extend
the knowledge about the models and to create new models on demand.
Mathematical analysis, developed in the 1 8th and 1 9th centuries to solve
dynamical problems in physics, consists of a series of models centered around
the real numbers and their functions. As examples, we mention continuous
functions, differentiable functions (of various orders), analytic functions, and
integrable functions; all classes of functions defined on various subsets of
euclidean space � n, and several classes also defined with vector values. Func­
tional analysis was developed in the first third of the 20th century by the
pioneering work of Banach, Hilbert, von Neumann, and Riesz, among others,
to establish a model for the models of analysis. Concentrating on "external"
properties of the classes of functions, these fit into a model that draws its

axioms from (linear) algebra and topology. The creation of such "super­
models" is not a new phenomenon in mathematics, and, under the name of
"generalization," it appears in every mathematical theory. But the users of
the original models (astronomers, physicists, engineers, et cetera) naturally
enough take a somewhat sceptical view of this development and complain
that the mathematicians now are doing mathematics for its own sake. As a
mathematician my reply must be that the abstraction process that goes into
functional analysis is necessary to survey and to master the enormous material
we have to handle. It is not obvious, for example, that a differential equation,


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viii

Preface

a system of linear equations, and a problem in the calculus of variations have
anything in common. A knowledge of operators on topological vector spaces
gives, however, a basis of reference, within which the concepts of kernels,
eigenvalues, and inverse transformations can be used on all three problems.
Our critics, especially those well-meaning pedagogues, should come to realize
that mathematics becomes simpler only through abstraction. The mathe­
matics that represented the conceptual limit for the minds of Newton and
Leibniz is taught regularly in our high schools, because we now have a clear
(i.e. abstract) notion of a function and of the real numbers.
When this defense has been put forward for official use, we may admit in
private that the wind is cold on the peaks of abstraction. The fact that the
objects and examples in functional analysis are themselves mathematical
theories makes communication with nonmathematicians almost hopeless and
deprives us of the feedback that makes mathematics more than an aesthetical

play with axioms. (Not that this aspect should be completely neglected.) The
dichotomy between the many small and directly applicable models and the
large, abstract supermodel cannot be explained away. Each must find his own
way between Scylla and Charybdis.
The material contained in this book falls under Kelley's label: What Every
Young Analyst Should Know. That the young person should know more (e.g.
more about topological vector spaces, distributions, and differential equa­
tions) does not invalidate the first commandment. The book is suitable for a
two-semester course at the first year graduate level. If time permits only a
one-semester course, then Chapters 1, 2, and 3 is a possible choice for its
content, although if the level of ambition is higher, 4. 1 -4.4 may be substituted
for 3.3-3.4. Whatever choice is made, there should be time for the student to
do some of the exercises attached to every section in the first four chapters.
The exercises vary in the extreme from routine calculations to small guided
research projects. The two last chapters may be regarded as huge appendices,
but with entirely different purposes. Chapter 5 on (the spectral theory of)
unbounded operators builds heavily upon the material contained in the
previous chapters and is an end in itself. Chapter 6 on integration theory
depends only on a few key results in the first three chapters (and may be
studied simultaneously with Chapters 2 and 3), but many of its results are
used implicitly (in Chapters 2-5) and explicitly (in Sections 4.5-4.7 and 5. 3 )
throughout the text.
This book grew out of a course on the Fundamentals of Functional
Analysis given at The University of Copenhagen in the fall of 1982 and again
in 1983. The primary aim is to give a concentrated survey of the tools of
modern analysis. Within each section there are only a few main results­
labeled theorems-and the remaining part of the material consists of sup­
porting lemmas, explanatory remarks, or propositions of secondary impor­
tance. The style of writing is of necessity compact, and the reader must be
prepared to supply minor details in some arguments. In principle, though, the

book is "self-contained." However, for convenience, a list of classic or estab-


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Preface

ix

lished textbooks, covering (parts of) the same material, has been added. In the
Bibliography the reader will also find a number of original papers, so that she
can judge for herself "wie es eigentlich gewesen."
Several of my colleagues and students have read (parts of) the manuscript
and offered valuable criticism. Special thanks are due to B. Fuglede, G. Grubb,
E. Kehlet, K.B. Laursen, and F. Tops0e.
The title of the book may convey the feeling that the message is urgent and
the medium indispensable. It may as well be construed as an abbreviation of
the scholarly accurate heading: Analysis based on Norms, Operators, and
Weak topologies.
Copenhagen

Gert Kjrergard Pedersen


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Contents

Preface

vii

CHAPTER 1
General Topology

1

1 . 1 . Ordered Sets

1

The axiom of choice, Zorn's lemma, and Cantors's well-ordering principle; and
their equivalence. Exercises.

1 .2. Topology

8

Open and closed sets. Interior points and boundary. Basis and subbasis for a
topology. Countability axioms. Exercises.

1 .3. Convergence

13

Nets and subnets. Convergence of nets. Accumulation points. Universal nets.
Exercises.

1 .4. Continuity

17

Continuous functions. Open maps and homeomorphisms. Initial topology.
Product topology. Final topology. Quotient topology. Exercises.

1.5. Separation

23

Hausdorff spaces. Normal spaces. Urysohn's lemma. Tietze's extension theorem.
Semicontinuity. Exercises.

1 .6. Compactness
Equivalent conditions for compactness. Normality of compact Hausdorff spaces.
Images of compact sets. Tychonoff's theorem. Compact subsets of 1Ii!". The
Tychonoff cube and metrization. Exercises.

30


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xii

Contents

1 .7. Local Compactness

36

One-point compactification. Continuous functions vanishing at infinity. Nor­
mality of locally compact, a-compact spaces. Paracompactness. Partition of
unity. Exercises.

CHAPTER 2
Banach Spaces

43

2. 1 . Normed Spaces

43

Normed spaces. Bounded operators. Quotient norm. Finite-dimensional spaces.
Completion. Examples. Sum and product of normed spaces. Exercises.

2.2. Category

52

The Baire category theorem. The open mapping theorem. The closed graph
theorem. The principle of uniform boundedness. Exercises.

2.3. Dual Spaces

56

The Hahn-Banach extension theorem. Spaces in duality. Adjoint operator.
Exercises.

2.4. Weak Topologies

62

Weak topology induced by seminorms. Weakly continuous functionals. The
Hahn-Banach separation theorem. The weak* topology. w*-c1osed subspaces
and their duality theory. Exercises.

2.5. w*-Compactness

69

Alaoglu's theorem. Krein-Milman's theorem. Examples of extremal sets. Extre­
mal probability measures. Krein-Smulian's theorem. Vector-valued integration.
Exercises.

CHAPTER 3
Hilbert Spaces

79

3 . 1 . Inner Products

79

Sesquilinear forms and inner products. Polarization identities and the Cauchy­
Schwarz inequality. Parallellogram law. Orthogonal sum. Orthogonal comple­
ment. Conjugate self-duality of Hilbert spaces. Weak topology. Orthonormal
basis. Orthonormalization. Isomorphism of Hilbert spaces. Exercises.

3.2. Operators on Hilbert Space

88

The correspondence between sesquilinear forms and operators. Adjoint operator
and involution in B(�). Invertibility, normality, and positivity in B(�). The
square root. Projections and diagonalizable operators. Unitary operators and
partial isometries. Polar decomposition. The Russo-Dye-Gardner theorem.
Numerical radius. Exercises.

3.3. Compact Operators
Equivalent characterizations of compact operators. The spectral theorem for
normal, compact operators. Atkinson's theorem. Fredholm operators and index.
Invariance properties of the index. Exercises.

105


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Contents

xiii

115

3.4. The Trace
Definition and invariance properties of the trace. The trace class operators and
the Hilbert-Schmidt operators. The dualities between Bo(f»), B'(f») and B(f»).
Fredholm equations. The Sturm-Liouville problem. Exercises.

CHAPTER 4
Spectral Theory

127

4. 1. Banach Algebras

128

Ideals and quotients. Unit and approximate units. Invertible elements.
C. Neumann's series. Spectrum and spectral radius. The spectral radius
formula. Mazur's theorem. Exercises.

4.2. The Gelfand Transform

1 37

Characters and maximal ideals. The Gelfand transform. Examples, including
Fourier transforms. Exercises.

4.3. Function Algebras

144

The Stone-Weierstrass theorem. Involution in Banach algebras. C*-algebras.
The characterization of commutative C*-algebras. Stone-Cech compactification
.
of Tychonoff spaces. Exercises.

4.4. The Spectral Theorem, I

1 56

Spectral theory with continuous function calculus. Spectrum versus eigenvalues.
Square root of a positive operator. The absolute value of an operator. Positive
and negative parts of a self-adjoint operator. Fuglede's theorem. Regular
equivalence of normal operators. Exercises.

4.5. The Spectral Theorem, II

1 62

Spectral theory with Borel function calculus. Spectral measures. Spectral
projections and eigenvalues. Exercises.

4.6. Operator Algebra

171

Strong and weak topology on B(f»). Characterization of stronglyjweakly contin­
uous functionals. The double commutant theorem. Von Neumann algebras.
The u-weak topology. The u-weakly continuous functionals. The predual
of a von Neumann algebra. Exercises.

4.7. Maximal Commutative Algebras

1 80

The condition III Ill'. Cyclic and separating vectors. .9'''''(X) as multiplication

operators. A measure-theoretic model for MA<;A's. Multiplicity-free operators.
MA<;A's as a generalization of orthonormal bases. The spectral theorem
revisited. Exercises.
=

CHAPTER 5
Unbounded Operators

191

5.1. Domains, Extensions, and Graphs

192

Densely defined operators. The adjoint operator. Symmetric and self-adjoint
operators. The operator T*T. Semibounded operators. The Friedrichs
extension. Examples.


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xiv

Contents

5.2. The Cayley Transform

203

The Cayley transform of a symmetric operator. The inverse transformation.
Defect indices. Affiliated operators. Spectrum of unbounded operators.

5.3. Unlimited Spectral Theory

209

Normal operators affiliated with a MAC;A. The multiplicity-free case. The
spectral theorem for an unbounded, self-adjoint operator. Stone's theorem.
The polar decomposition.

CHAPTER 6
Integration Theory

221

6. 1 . Radon Integrals

221

Upper and lower integral. Daniell's extension theorem. The vector lattice !e1(X).
Lebesgue's theorems on monotone and dominated convergence. Stieltjes integrals.

6.2. Measurability

228

Sequentially complete function classes. a-rings and a-algebras. Borel sets and
functions. Measurable sets and functions. Integrability of measurable functions.

6.3. Measures

235

Radon measures. Inner and outer regularity. The Riesz representation theorem.
Essential integral. The a-compact case. Extended integrability.

6.4. LP-spaces

239

Null functions and the almost everywhere terminology. The HOlder and
Minkowski inequalities. Egoroff's theorem. Lusin's theorem. The Riesz-Fischer
theorem. Approximation by continuous functions. Complex spaces. Interpola­
tion between !eP-spaces.

6.5. Duality Theory

247

a-compactness and a-finiteness. Absolute continuity. The Radon-Nikodym
theorem. Radon charges. Total variation. The Jordan decomposition. The
duality between LP-spaces.

6.6. Product Integrals

255

Product integral. Fubini's theorem. Tonelli's theorem. Locally compact groups.
Uniqueness of the Haar integral. The modular function. The convolution
algebras LI(G) and M(G).

Bibliography
List of Symbols
Index

267
27 1
273


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CHAPTER 1

General Topology

General or set-theoretical topology is the theory of continuity and conver­
gence in analysis. Although the theory draws its notions and fundamental
examples from geometry (so that the reader is advised always to think of a
topological space as something resembling the euclidean plane), it applies
most often to infinite-dimensional spaces of functions, for which geometrical
intuition is very hard to obtain. Topology allows us to reason in these situa­
tions as if the spaces were the familier two- and three-dimensional objects, but
the process takes a little time to get used to.
The material presented in this chapter centers around a few fundamental
topics. For example, we only introduce Hausdorff and normal spaces when
separation is discussed, although the literature operates with a hierarchy of
more than five distinct classes. A mildly unusual feature in the presentation
is the central role played by universal nets. Admittedly they are not easy to
get aquainted with, but they facilitate a number of arguments later on (giving,
for example, a five-line proof of Tychonoff's theorem). Since universal nets
entail the blatant use of the axiom of choice, we have included (in the regie

of naive set theory) a short proof of the equivalence among the axiom of
choice, Zorn's lemma, and Cantor's well-ordering principle. All other topics
from set theory, like ordinal and cardinal numbers, have been banned to the
exercise sections. A fate they share with a large number of interesting topo­
logical concepts.
1 . 1 . Ordered Sets

Synopsis. The axiom of choice, Zorn's lemma, and Cantor's well-ordering
principle, and their equivalence. Exercises.


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1 . General Topology

2

1.1.1. A binary relation in a set X is just a subset R of X x X. It is customary,
though, to use a relation sign, such as � , to indicate the relation. Thus
(x, y) E R is written x � y.
An order in X is a binary relation, written � , which is transitive (x � Y and
Y � z implies that x � z), reflexive (x � x for every x), and antisymmetric
(x � y and y � x implies x = y). We say that (X, � ) is an ordered set. Without
the anti symmetry condition we have a preorder, and much of what follows
will make sense also for preordered sets.
An element x is called a majorant for a subset Y of X, if y � x for every y
in Y. Minorants are defined analogously. We say that an order is filtering
upward, if every pair in X (and, hence, every finite subset of X) has a majorant.
Orders that are filtering downward are defined analogously. If a pair x, y in
X has a smallest majorant, relative to the order � , this element is denoted
x v y. Analogously, x A y denotes the largest minorant of the pair x, y, if it

exists. We say that (X, � ) is a lattice, if x v y and x A y exist for every pair
x, y in X. Furthermore, (X, � ) is said to be totally ordered if either x � y or
y � x for every pair x, y in X. Finally, we say that (X, � ) is well-ordered if
every nonempty subset Y of X has a smallest element (a minorant for Y
belonging to Y). This element we call the first element in Y.
Note that a well-ordered set is totally ordered (put Y = {x, y} ), that a totally
ordered set is a (trivial) lattice, and that a lattice order is both upward and
downward filtering. Note also that to each order � corresponds a reverse
order � , defined by x � y iff y � x.
1.1.2. Examples of orderings are found in the number systems, with their usual
orders. Thus, the set N of natural numbers is an example of a well-ordered
set. (Apart from simple repetitions, NuNu . . . , this is also the only concrete
example we can write down, despite 1 . 1 .6.) The sets Z and � are totally
ordered, but not well-ordered. The sets Z x Z and � x � are lattices, but not
totally ordered, when we use the product order, i.e. (Xl , X2) � (Yl , Yl) when­
ever Xl � Yl and X2 � Y2' [If, instead, we use the lexicographic order, i.e.
(Xl , X 2 ) � (Yl , Y2) if either Xl < Yl ' or Xl = Yl and X2 � Y2 ' then the sets
become totally ordered.]
An important order on the system 9'(X) of subsets of a given set X is given
by inclusion; thus A � B if A c B. The inclusion order turns 9'(X) into a
lattice with 0 as first and X as last elements. In applications it is usually the
reverse inclusion order that is used, i.e. A � B if A :::> B. For example, taking
X to be a sequence (xn) of real numbers converging to some x, and putting
T,. = {xk l k � n}, then clearly it is the reverse inclusion order on the tails T,.
that describe the convergence of (xn) to X.
1.1.3. The axiom of choice, formulated by Zermelo in 1904, states that for each
nonempty set X there is a (choice) function

c: 9'(X)\ {0} � X,

satisfying c(Y) E Y for every Y in 9'(X)\ {0}.


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3

1 . 1 . Ordered Sets

Using this axiom·Zerrnelo was able to give a satisfactory proof of Cantor's
well-ordering principle, which says that every set X has an order � , such that
(X, � ) is well-ordered.
The well-ordering principle is a necessary tool in proofs "by induction,"
when the set over which we induce is not a segment of 1\1 (so-called transfinite
induction). More recently, these proofs have been replaced by variations that
pass through the following axiom, known in the literature as Zorn's lemma
(Zorn 1935, but used by Kuratowski in 1922). Let us say that (X, � ) is
inductively ordered if each totally ordered subset of X (in the order induced
from X), has a majorant in X. Zorn's lemma then states that every inductively
ordered set has a maximal element (i.e. an element with no proper majorants).
1.1.4. Let (X, � ) be an ordered set and assume that c is a choice function for
X. For any subset Y of X, let maj ( Y) and min( Y), respectively, denote the sets
of proper majorants and minorants for Y in X. Thus x E maj ( Y) if y < x for
every y in Y, where the symbol y < x of course means y � x and y i= x.
A subset C of X is called a chain if it is well-ordered (relative to � ) and if
for each x in C we have

c(maj(C n min {x} » = x.
Note that c(X) is the first element in any chain and that { c (X)} is a chain
(though short).
1.1.5. Lemma. If C l and Cz are chains in X such that C l cf: Cz , there is an

element X l in Cl such that

PROOF. Since Cl \ Cz i= 0 and Cl is well-ordered, there is a first element X l in

C l \ Cz . By definition we therefore have
Cl n min {xd
(i)

c

Cz .

If the inclusion in (i) is proper, the set Cz \(C l n min {x l } ) has a first element
xz, since Cz is well-ordered. By definition, therefore,
(ii)
If the inclusion in (ii) is proper, the set (Cl n min {xd)\min {xz } (contained
in Cl n Cz ) has a first element y. By definition
(iii)

However, if y � x for some x in Cz n min {xz }, then y E Cz n min {xz }, con­
tradicting the choice of y. Since both x and y belong to the well-ordered, hence
totally ordered, set Cz , it follows that x < y for every x in Cz n min {xz }.
Thus in (iii) we actually have equality. Since both C l and Cz are chains
(relative to the same ordering and the same choice function), it follows from
the chain condition (*) in 1 . 1 .4 that y = xz . But Y E C l n min {xd while


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1 . General Topology

4

X 2 ¢ Cl n min { xd. To avoid a contradiction we must have equality in
(ii). Applying the chain condition to (ii) gives X l = X 2 in contradiction with
X l ¢ C2 and X 2 E C2 • Consequently, we have equality in (i), which is the desired
result.
[]
1.1.6. Theorem. The following three propositions are equivalent:

(i) The axiom of choice.
(ii) Zorn's lemma.
(iii) The well-ordering principle.
PROOF (i) => (ii). Suppose that (X, �) is inductively ordered, and by assumption

let c be a choice function for X. Consider the set {CN E J} of all chains in X
and put C = U Cj• We claim that for any X in Cj we have

C n min {x} = Cj n min {x}.
For if y belongs to the first (obviously larger) set, then y E Ci for some i in J.
Either Ci c Cj ' in which case y E Cj ' or Ci cj: Cj . In that case there is by 1 . 1 .5 an
X i in Ci such that Cj = Ci n min { x J As y < X < Xi ' we again see that y E Cj .
It now follows easily that C is well-ordered. For if 0 i= Y c C, there is a j
in J with Cj n Y i= 0. Taking y to be the first element in Cj n Y it follows from
(**) that y is the first element in all of Y. Condition (**) also immediately shows
that C satisfies the chain condition (*) in 1 . 1 .4. Thus C is a chain, and it is
clearly the longest possible. Therefore, maj (C) = 0. Otherwise we could take
X o = c(maj (C» E maj (C),
and then C u { xo } would be a chain [(*) in 1 . 1 .4 has just been satisfied for xo]
effectively longer than C.
Since the order is inductive, the set C has a majorant x., in X. Since

maj(C) = 0, we must have x., E C, i.e. x., is the largest element in C. But then
x., is a maximal element in X, because any proper majorant for x., would
belong to maj(C).
(ii) => (iii). Given a set X consider the system M of well-ordered, nonempty
subsets (Cj , �j ) of X. Note that M i= 0, the one-point sets are trivial members.
We define an order � on M by setting (Ci, �i) � (Cj , �j ) if either Ci = Cj and
� i = �j' or if there is an Xj in Cj such that
(***)
Ci = { x E Cjl x �j Xj} and �i = �jI Ci '
The claim now is that (M, �) is inductively ordered. To prove this, let N be a
totally ordered subset of M and let C be the union of all Cj in N. Define � on
C by X � Y whenever { x , y } c Cj E N and X �j Y ' Note that if { x, y } c Ci E N,
then X �i Y iff X �j Y bec�use of the total ordering of N, so that � is a
well-defined order on C. Exactly as in the proof of (i) => (ii) one shows that if
X E Cj, then
C n min {x} = Cj n min {x}

(**)


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1 . 1. Ordered Sets

5

(the result of 1 . 1 .5 has been built into the order on M). As before, this implies
that (C, ::;; ) is well-ordered. The conclusion that (C, ::;; ) is a majorant for N is
trivial if N has a largest element (which then must be C). Otherwise, each
(Cj, ::;; j) has a majorant (Cj , ::;;j ) in N and is thus of the form (***) relative to
Cj ; and, as (**) shows, also of the form (***) relative to C. We conclude that

(C, ::;; ) is a majorant for N, which proves that M is inductively ordered.
Condition (ii) now implies that M has a maximal element (X." ::;; ., ). If
X., =1= X, we choose some x., in X\X., and extend ::;; ., to X., u {X., } by setting
x ::;;., x., for every x in X.,. This gives a well-ordered set (X., u {X.,}, ::;; ., ) that
majorizes (X." ::;; ., ) in the ordering in M, contradicting the maximality of
(X." ::;; ., ). Thus X = X., and is consequently well-ordered.
(iii) => (i). Given a nonempty set X, choose a well-order ::;; on it. Now define
c(Y) to be the first element in Y for every nonempty subset Y of X.
D
1.1.7. Remark. The subsequent presentation in this book builds on the ac­
ceptance of the axiom of choice and its equivalent forms given in 1 . 1 .6. In the
intuitive treatment of set theory used here, according to which a set is a
properly determined collection of elements, it is not possible precisely to
explain the role of the axiom of choice. For this we would need an axiomatic
description of set theory, first given by Zermelo and Fraenkel. In 1 938 G6del
showed that if the Zermelo-Fraenkel system of axioms is consistent (that in
itself an unsolved question), then the axiom of choice may be added without
violating consistency. In 1963 Cohen showed further that the axiom of choice
is independent of the Zermelo-Fraenkel axioms. This means that our accep­
tance of the axiom of choice determines what sort of mathematics we want to
oreate, and it may in the end affect our mathematical description of physical
realities. The same is true (albeit on a smaller scale) with the parallel axiom
in euclidean geometry. But as the advocates of the axiom of choice, among
them Hilbert and von Neumann, point out, several key results in modern
mathematical analysis [e.g. the Tychonoff theorem (1.6. 10), the Hahn-Banach
theorem (2.3.3), the Krein-Milman theorem (2.5.4), and Gelfand theory (4.2.3)]
depend crucially on the axiom of choice. Rejecting it, one therefore loses a
substantial part of mathematics, and, more important, there seems to be no
compensation for the abstinence.

EXERCISES
E 1.1.1. A subset 5l of a real vector space X is called a cone if 5l + 5l c 5l
and 1R +5l = R If in addition -5l ('\ 5l = {O} and 5l - 5l = X, we say
that 5l generates X . Show that the relation in X defined by x ::;; y if
y x E 5l is an order on X if 5l is a generating cone. Find the set
{x E Xix � O}, and discuss the relations between the order and the
-

vector space structure. Find the condition on 5l that makes the order
total. Describe some cones in IR" for n = 1, 2, 3.


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1 . General Topology

6

E 1.1.2. Let oc be a positive, irrational number and show that the relation in
lL x lL given by

(X l , X 2 ) � ( Y 1 , Y 2 ) if OC( Y 1 - xd � Y 2 - X 2
is a total order. Sketch the set
E 1.1.3. An order isomorphism between two ordered sets (X, � ) and (Y, � ) is

a bijective map q>: X -+ Y such that X l � x 2 iff q>(xd � q>(X 2 ). A
segment of a well-ordered set (X, � ) is a subset of X of the form
min {x} for some X in X, or X itself (the improper segment). Show
that if X and Y are well-ordered sets, then either X is order iso­
morphic to a segment of Y (with the relative order) or Y is order
isomorphic to a segment of X.

Hint: The system of order isomorphisms q>: X", -+ Y"" where X",
and Y", are segments of X and Y, respectively, is inductively ordered
if we define q> � 1/1 to mean X", c X", (which implies that q> = 1/1 I X'"
and thus Y", c Y",). Prove that for a maximal element q>: X", -+ Y",
either X", or Y", must be an improper segment.

E 1.1 .4. The equivalence classes of well-ordered sets modulo order isomor­

phism (E 1 . 1 .3) are called ordinal numbers. Every well-ordered set has
thus been assigned a "size" determined by its ordinal number. Show
that the class of ordinal numbers is well-ordered.
Hint: Given a collection of ordinal numbers {ocN E J} choose a
corresponding family of well-ordered sets (XN E J} such that OCj is
the ordinal number for Xj for every j in J. Now fix one Xj . Either its
equivalence class ocj is the smallest (and we are done) or each one of
the smaller X;'s is order isomorphic to a proper segment min {Xi} in
Xj by E 1 . 1.3. But these segments form a well-ordered set.

E 1.1.5. Let f: X -+ Y and g: Y -+ X be injective (but not necessarily surjec­

tive) maps between the two sets X and Y. Show that there is a
bijective map h: X -+ Y (F. Bernstein, 1 897).
Hint: Define
00

A = U (g 0 f)ft(X\g( Y»,
ft= O
and put h = f on A and h = g- l on X\A. Note that X\g(Y) c A,
whereas Y\f(A) c g- l (X\A).
E 1.1.6. We define an equivalence relation on the class of sets by setting

X '" Y if there exists a bijective map h: X -+ Y. Each equivalence
class is called a cardinal number. Show that the natural numbers are
the cardinal numbers for finite sets. Discuss the "cardinality" of some
infinite sets, e.g. N, lL, IR, and 1R 2 .


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1.1. Ordered Sets

7

E 1.1.7. For the cardinal numbers, defined in E 1 . 1 .6, we define a relation
by letting oc � P if there are sets A and B with card(A) = oc and
card(B) = p (a more correct, but less used terminology would be
A E oc and B E p, cf. E 1 . 1 .6), and an injective map f: A --+ B. Show

with the use of E 1 . 1.5 that � is an order on the cardinal numbers.
Show finally that the class of cardinal numbers is well-ordered.
Hint: If {XN E J} is a collection of sets with corresponding car­
dinal numbers ocj = card(Xj ), we can well-order each Xj ' and then
apply E 1 . 1 .4.

E.l.1.S. A set is called countable (or countably infinite) if it has the same

cardinality (cf. E 1 . 1.6) as the set 1\1 of natural numbers. Show that
there is a well-ordered set (X, � ), which is itself uncountable, but
which has the property that each segment min {x} is countable if
X E X.
Hint: Choose a well-ordered set ( Y, � ) that is uncountable. The

subset Z of elements z in Y such that the segment min {z} is un­
countable is either empty (and we are done) or else has a first element
n. Set X = min {n}. The ordinal number (corresponding to) n is
called the first uncountable ordinal.

E.1.1.9. Let X be a vector space over a field IF. A basis for X is a subset
� = { eN E J} of linearly independent vectors from X, such that
every x in X has a (necessarily unique) decomposition as a finite
linear combination of vectors from �. Show that every vector space

has a basis.
Hint: A basis is a maximal element in the system of linearly
independent subsets of X .

E 1.1.10. Show that there exists a discontinuous function f: IR --+ IR, such that
f(x + y) = f(x) + f(y) for all real numbers x and y. Show that f(l)

contains arbitrarily (numerically) large numbers for every (small)
interval I in IR.
Hint: Let (!) denote the field of rational numbers and apply E 1 . 1 .9
with X = IR and IF = (!) to obtain what is called a Hamel basis for
IR. Show that f can be assigned arbitrary values on the Hamel
basis and still have an (unique) extension to an additive function
on IR.

E.l.1.11. Let X be a set and 9'(X) the family of all subsets of X. Show that

the cardinality of the set 9'(X) is strictly larger than that of X, cf.
E 1 . 1 .6.
Hint: If f: X --+ 9'(X) is a bijective function, set

A = {x E X l x!f f(x)},

and take y = f- l (A). Either possibility Y E A or Y!f A will lead to
a contradiction.


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8

1 . General Topology

1 .2. Topology

Synopsis. Open and closed sets. Interior points and boundary. Basis and
subbasis for a topology. Countability axioms. Exercises.
1.2.1. A topology on a set X is a system t of subsets of X with the properties that

(i) every union of sets in t belongs to t.
(ii) every finite intersection of sets in t belongs to t.
(iii) 0 E t and X E t.
We say that (X, t) is a topological space, and that t consists of the open subsets
in (X, t) .
1.2.2. A metric on a set X is a function d: X x X ..... � + (the distance function)
that is symmetric [d(x, y) = d(y, x)] and faithful [d(x, y) = 0 iff x = y], and
satisfies the triangle inequality [d(x, y) ::;;; d(x, z) + d(z, y)]. We declare a subset
A of X to be open if for each x in A, there is a sufficiently small a > 0, such

that the a-ball {y E X l d (x, y) < a} around x is contained in A. It is straight­
forward to check that the collection of such open sets satisfies the requirements
(i)-(iii) in 1.2. 1 , and thus gives a topology on X, the induced topology. Con­

versely, we say that a topological space (X, t) is metrizable if there is a metric
on X that induces t.
1.2.3. Remark. It is a fact that the overwhelming number of topological spaces
used in the applications are metrizable. The question therefore arises: What
is topology good for? The answer is (hopefully) contained in this chapter, but
a few suggestions can be given already now: Using topological rather than
metric terminology, the fundamental concepts of analysis, such as conver­
gence, continuity, and compactness, have simple formulations, and the argu­
ments involving them become more transparent. As a concrete example, con­
sider the open interval ] - 1, 1 [ and the real axis �. These sets are topologically
indistinguishable [the map x ..... tan(!nx) furnishes a bijective correspondence
between the open sets in the two spaces]. This explains why every property
of � that only depends on the topology also is found in ] - 1, 1 [. Metrically,
however, the spaces are quite different. (� is unbounded and complete; ] - 1 , 1 [
enjoys the opposite properties.) A metric o n a topological space may thus
emphasize certain characteristics that are topologically irrelevant.
1.2.4. A subset Y in a topological space (X, t) is a neighborhood of a point x
in X if there is an open set A such that x E A c Y. The system of neighborhoods
of x is called the neighborhood filter and is denoted by lD(x). The concept
of neighborhood is fundamental in the theory, as the name topology in­
dicates (topos = place; logos = knowledge). A rival name (now obsolete) for
the theory was analysis situs, which again stresses the importance of neighbor­
hoods (situs = site).


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9

1.2. Topology

A point x is an inner point in a subset Y of X if there is an open set A such
that x E A c Y. The set of inner points in Y is denoted by yo. Note that yo is
the set of points for which Y is a neighborhood, and yo is the largest open set
contained in Y.
1.2.5. A subset F of a topological space (X, t) is closed if X\ F E t. The definition
implies that 0 and X are closed sets and that an arbitrary intersection and a
finite union of closed sets is again closed.
For each subset Y of X we now define the closure of Y as the intersection
Y- of all closed sets containing Y. The elements in Y - are called limit points
for Y. Note the formulas

X\ YO = (X\ Yf ·

X\ Y- = (X\ y)0,

We say that a set Y is dense in a (usually larger) set Z, if Z c Y -.
1.2.6. Proposition. If Y c X and x E X, then x E Y- iff Y n A oF 0 for each A

in (9(x) .

(0)')
� without loss of generality we

PROOF. If Y n A = 0 for some A in (9(x

may assume that A E t. Thus X\ A is a closed set containing Y, so that
Y- c X\A and x rf; Y-.
Conversely, if x rf; Y-, then X\ Y- is an open neighborhood of x disjoint
�Y.
D

1.2.7. For Y c X the set Y- \ yo is called the boundary of Y and is denoted by
oY. We see from 1 .2.6 that x E o Y iff every neighborhood of x meets both Y
and X\ Y. In particular,o Y = o(X\ Y). Note that a closed set contains its
boundary, whereas an open set is disjoint from its boundary.
1.2.S. If (X, t) is a topological space we define the relative topology on any
subset Y of X to be the collection of sets of the form A n Y, A E t. It follows
that a subset of Y is closed in the relative topology iff it has the form Y n F
for some closed set F in X. To avoid ambiguity we shall refer to the relevant
subsets of Y as being relatively open and relatively closed.
1.2.9. If (J and t are two topologies on a set X, we say that (J is weaker than t
or that t is stronger than (J, provided that (J c t. This defines an order on the
set of topologies on X. There is a first element in this ordering, namely the
trivial topology, that consists only of the two sets 0 and X. There is also a last
element, the discrete topology, containing every subset of X. As the next result
shows, the order is a lattice in a very complete sense.
1.2.10. Proposition. Given a system { tN E J } of topologies on a set X, there is
a weakest topology stronger than every tj ' and there is a strongest topology
weaker than every tj . These topologies are denoted v tj and A tj , respectively.


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10

1. General Topology

PROOF. Define 1\ tj as the collection of subsets A in X such that A E tj for all
j in J. This is a topology, and it is weaker than every tj , but only minimally so.
Now let T denote the set of topologies on X that are stronger than every
tj . The discrete topology belongs to T, so T i= 0. Setting v tj = 1\ t, t E T, we
obtain a topology with the required property.

0
1.2.11. Given any system p of subsets of X there is a weakest topology t(p)
that contains p, namely, t(p) = 1\ t, where t ranges over all topologies on X
that contain p. We say that p is a subbasis for t(p). If each set in t(p) is a union
of sets from p, we say that p is a basis for t(p). It follows from 1.2. 1 2 that this
will happen iff each finite intersection of sets from p is the union of sets from
p. In particular, p is a basis for t(p) if it is stable under finite intersections.
Given a topological space (X, t), we say that a system p of subsets of X
contains a neighborhood basis for a point x in X, if for each A in (9(x), there
is a B in p n (9(x), such that B c A. The reason for this terminology becomes
clear from the next result.
1.2.12. Proposition. For a system p of subsets of X, the topology t(p) consists
of exactly those sets that are unions of sets, each of which is a finite intersection
of sets from p, together with 0 and X.
Conversely, a system p of open sets in a topological space (X, t) is a basis
(respectively, a subbasis) for t, if p (respectively, the system offinite intersections
of sets from p) contains a neighborhood basis for every point in X.

PROOF. The system of sets described in the first half of the proposition is
stable under finite intersections and arbitrary unions, and it contains 0 and
X (per fiat). It therefore is a topology, and clearly the weakest one that
contains p.
Conversely, if a system pet contains (or after taking finite intersections
contains) a neighborhood basis for every point, let t(p) be the topology it
generates and note that t(p) c t. If A E t, there is for each x in A a B(x) in p
[respectively, in t(p)] such that x E B(x) c A. Since A = U B(x), we see that
p is a basis for t [respectively, t = t(p)].
0
1.2.13. A topological space (X, t) is separable if some sequence of points is
dense in X.

A topological space (X, t) satisfies the first axiom of countability if for each
x in X there is a sequence (An(x)) in (9(x), such that every A in (9(x) contains
some An(x) (i.e. if every neighborhood filter has a countable basis).
A tdpological space (X, t) satisfies the second axiom of countability if t has
a countable basis. According to 1 .2. 1 2 it suffices for t to have a countable
subbasis, because finite intersections of subbasis sets will then be a countable
basis.
The three conditions mentioned above all say something about the "size"
of t, and the second countability axiom (which implies the two previous