Methods of modern mathematical physics volume 1 functional analysis michael reed, barry simon

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I:

FUNCTIONAL ANALYSIS

Revised and Enlarged Edition

MICHAEL REED

BA RRY SIMON

Department of Mathematíes
Duke University

Departments of Mal hematíes
and Physics
Princeton University

ACADEMIC P R E SS , INC.
Harcourt Brace Jovanovlch, Publlshera

San Diego
london

New York

Sydney

Berketey

Tokyo

Toronto

Boston

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METHODS OF
MODERN MATHEMATICAL PHYSICS


To
R . S . P hillip s a n d A . S . W ightm an,

C o p y r i g h t © 1980, b y A c a d e m i c P r e s s , I n c .
ALL RIGHTS RESERVED.
NO PART OF TH IS P U B U C A T IO N MAY BE REPRO DU CED OR
TRA N SM ITTED IN ANY F O R M OR BY ANY M EA N S, EL EC TR O N IC
OR M ECHANICAL, INCLUDJNG PH O TO C O PY , RECORDING, OR ANY
IN FO R M ATION STORAGE AND RETRIEVAL SY STE M , W IT H O U T
PER M ISSIO N IN W R IT IN G FR O M TH E PUBL1SHER.

ACADEMIC PRESS, INC.
1 2 5 0 S ix t h A v e n u e , S an D i e g o , C a lif or n ia 9 2 1 0 1

United Kingdom Edition published by
ACADEMIC PRESS, INC. (LO N D O N ) LTD.

2 4 /2 8 O v al R o a d , L o n d o n N W 1

7D X

Library of Congress Cataloging in Publication Data
Reed, Michael.
Methods of modern mathematical physics.
Vol. 1 Functional analysis, revised and enlarged edition.
Includes bibliographical references.
CONTENTS: v. 1. Functional analysis.-v. 2. Fourier
analysis, self-adjointness.-v. 3. Scattering theory.-v. 4.
Analysis of operators.
1. Mathematical physics. I. Simón, Barry.joint
author. II. Title.
QC20.R37 1972
530.1’5
75-182650
ISBN 0 -1 2 - 5 8 5 0 5 0 -6 (v. 1)
AMS (MOS) 1970 Subject Classifications: 4 6 -0 2 ,4 7 - 0 2 , 4 2 -0 2

P R I N T E D IN T H E U N I T E D S T A T E S O F A M E R I C A
8 8 89 9 0 91 92

10 9 8 7 6 5 4

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M entors, C olleagues, F rien d s

This book is the first o f a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics. It describes the funda­
mental principies o f functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few
applications when we thought that they would provide motivation for the reader.
Later volumes describe various advanced topics in functional analysis and give
numerous applications in classical physics, modem physics, and partial differential equations.
This revised and enlarged edition differs from the first in two major ways.
First, many coileagues have suggested to us that it would be helpful to include
some material on the Fourier transform in Volume I so that this important topic
can be conveniently included in a standard functional analysis course using this
book. Thus, we have included in this edition Sections IX. 1, IX .2, and part of
IX .3 from Volume II and some additional material, together with relevant notes
and problems. Secondly, we have included a variety o f supplementary material
at the end o f the book. Some o f these supplementary sections provide proofs of
theorems in Chapters II - I V which were omitted in the first edition. W hile these
proofs make Chapters I I - I V more self-contained, we still recommend that students with no previous experience with this material consult more elementary
texts. Other supplementary sections provide expository material to aid the in­
structor and the student (for example, “ Applications of Compact O perators” ).
Still other sections introduce and develop new material (for example, “ Minimization o f Functionals ” ).
It gives us pleasure to thank many individuáis:
The students who took our course in 1970-1971 and especially J. E. Taylor
for constructive comments about the lectures and lecture notes.
L. Gross, T. Kato, and especially D. Ruelle for reading parts o f the manuscript and for making numerous suggestions and corrections.

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P reface


F. Armstrong, E. Epstein, B. Farrell, and H. Wertz for excellent typing.
M. Goldberger, E. Nelson, M. Simón, E. Stein, and A. Wightman for aid

and encouragement.
M ike R eed
B arry S imón

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April 1980


M athem atics has its roots in num eroiogy, geometry, and physics. Since the
tim e o f N ew ton, the search for m athem atical m odels for physical phenom ena
has been a source o f m athem atical problem s. In fact, whole branches o f
m athem atics have grown out o f attem pts to analyze particular physical
situations. An example is the developm ení o f harm onic analysis from F ourier’s
work on the heat equation.
A lthough m athem atics and physics have grown ap art in this century,
physics has continued to stim ulate m athem atical research. Partially because
o f this, the influence o f physics on m athem atics is well understood. However,
the contributions o f m athem atics to physics are not as well understood. Jt is
a com m on fallacy to suppose th a t m athem atics is im postant for physics only
because it is a useful tool for m aking com putations. Actually, m athem atics
plays a m ore subtle role which in the long run is m ore im portant. W hen a
successful m athem atical m odel is created for a physical phenom enon, that is,
a model which can be used for accurate com putations and predictions, the
m athem atical structure o f the model itself provides a new way o f thinking
about the phenom enon. Put slightly differently, when a model is successful
it is natural to think o f the physical quantities in term s o f the m athem atical
objects which represent them and to interpret sim ilar o r secondary phenom ena
in term s o f th e sam e model. Because o f this, an investigation o f the internal
m athem atical structure o f the model can alter and enlarge our understanding

o f the physical phenom enon. O f course, th e outstanding exam ple o f this is
N ew tonian m echanics which provided such a clear and coherent picture o f
celestial m otions th at it was used to interpret practically all physical
phenom ena. The model itself becam e central to an understanding o f the
physical world and it was difficult to give it up in the late nineteenth century,
even in the face o f contradictory evidence. A m ore m odern exam ple o f this
influence o f m athem atics on physics is the use o f group theory to classify
elem entary particles.
vii

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Introduction


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The analysis o f m athem atical models for physical phenom ena is p art o f
the subject m atter o f m athem atical physics. By analysis is m eant both the
rigorous derivation o f explicit form ulas and investigations of the internal
m athem atical structure o f the models. In both cases the m athem atical problems which arise iead to m ore general m athem atical questions n ot associated
with any particular model. A lthough these general questions are som etim es
problem s in puré m athem atics, they are usually classified as m athem atical
physics since they arise from problem s in physics.
M athem atical physics has traditionally been concerned with the m athe­
m atics o f classical physics: m echanics, fluid dynam ics, acoustics, potential
theory, and optics. The m ain m athem atical too! for the study o f these
branches o f physics is the theory of ordinary and partiai differential equations
and related areas like integral equations and the calculus o f variations. This
classical m athem atical physics has long been part o f curricula in m athem atics

and physics departm ents. However, since 1926 the frontiers o f physics have
been concentrated increasingly in quantum mechanics and the subjects opened
up by the quantum theory: atom ic physics, nuclear physics, solid State
physics, elem entary particle physics. The central m athem atical discipline for
the study o f these branches o f physics is functional analysis, though the
theories o f group representations and several complex variables are also
im portant. Von N eum ann began the analysis o f the fram ew ork o f quantum
mechanics in the years foliowing 1926, but there were few attem pts to study
the structure o f specific quantum systems (exceptions would be some o f the
work o f Friedrichs and Rellich). This situation changed in the early 1950’s
when K ato proved the self-adjointness o f atom ic H am iltonians and G árding
and W ightm an form ulated the axioms for quantum field theory. These events
dem onstrated the usefulness o f functional analysis and pointed out the m any
difficult m athem atical questions arising in m odern physics. Since then the
range and breadth o f both the functional analysis techniques used and the
subjects discussed in m odern m athem atical physics have increased enorm ously.
The problem s range from the concrete, for exam ple how to com pute or
estím ate the point spectrum o f a particular operator, to the general, for
example the rep resen taro n theory o f C “"-algebras. The techniques used and
the general approach to the subject have becom e m ore abstract. A lthough
in some areas the physics is so well understood th at the problem s are exercises
in puré m athem atics, there are other areas where neither the physics ñor the
m athem atical models are well understood. These developm ents have had
severa! serious effects not the least o f which is the difficulty o f com m unication
between m athem aticians and physicists. Physicists are often dismayed at the
breadth o f background and increasing m athem atical sophistication which are
required to understand the models. M athem aticians are often frastrated by


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their own inability to undersíand the physics and the inability o f physicists
to form úlate the problem s in a way th at m athem aticians can understand.
A few specific rem arks are appropriate. The prerequisite for reading this
volume is roughly the m athem atical sophistication acquired in a typical
undergraduate m athem atics education in the U nited States. C hapter I is
intended as a review o f b ackground m aterial. W e expect th a t the reader will
have som e acquaintance w ith parts o f the m aterial covered in C hapters II—IV
and have occasionally om itted proofs in these chapters when they seem
uninspiring and unim portant for the reader.
The m aterial in this book is sufficient for a two-semester course. A lthough
we taught m ost o f the m aterial in a special one-semester course at Princeton
which m et five days a week, we d o not recom m end a repetition o f that, either
for facuity or students. In order th at the m aterial may be easily adapted for
lectures, we have w ritten most o f the chapters so that the earlier sections
contain the basic topics while the iater sections contain more specialized and
advanced topics and applications. F o r example, one can give students the
basic ideas about unbounded operators in nine or ten lectures from
Sections 1-4 o f C hapter VIII. On the other hand, by doing the details of
the proofs and by adding m aterial from the notes and problem s, C hapter VIII
could easily become a one-sem ester course by itself.
Each chapter o f this book ends with a long set o f problem s. Some o f the
problem s fill gaps in the text (these are m arked with a dagger). O thers develop
altérnate proofs to the theorem s in the text or introduce new material. We
have also included harder problem s (indicated by a star) in order to challenge
the reader. W e strongly encourage students to do the problem s. It is trite but
true th a t m athem atics is learned by doing it, not by w atching other
people do it.
We hope th at these volumes will provide physicists with an access to
m odern abstract techniques and th at m athem aticians will benefit by learning

the advanced techniques side by side with their applications.


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Preface
Introductioti
Contents o f Other Voluntes

I:

v
vii
xv

PRELIMINARES
!. Sets and functions
2. Metric and normed linear spaces
Appendix
Lint sup and lim in f
3. The Lebesgue integral
4. Abstract measure theory
5. Two convergence arguments
6. Equicontinuity
Notes
Problems

II:

l
3
11
12
19
26
28
31
32

HILBERT SPACES
/ . The geometry o f Hilbert space
2. The Riesz lemma
3. Orthonormal bases
4. Tensor producís o f Hilbert spaces
5. Ergodic theory: an introduction
Notes
Problems

36
41
44
49
54
60
63
xi

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Contents


BANACH SPACES
7. Definition and examples
2. Duals and double duals
3. The H ahn-B anach theorem
4. Operations on Banach spaces
5. The Baire category theorem and its consequences
Notes
Problems

67
72
75
78
79
84
86

IV: TOPOLOGICAL SPACES
1. General notions
2. Nets and convergence
3. Compactness
Appendix The Stone-W eierstrass theorem
4. Measure theory on compact spaces
5. Weak topologies on Banach spaces
Appendix Weak and strong measurability
Notes
Problems

V:

90
95
97
103
¡04
111
115
117
119

LOCALLY CONVEX SPACES
/. General properties
2. Fréchet spaces
3. Functions o f rapid decease and the tempered distribuíions
Appendix The N-representation fo r S f and 9 ”
4.

¡ndu ctive lim its: g e n e ra lize d fu n ctio n s a n d w eak Solutions o f
p a r d a l d ifferen tia l eq u a tio n s

5. Fixed point theorems
6. Applications o f fix e d point theorems
7. Topologies on locally convex spaces: duality theory and the
strong dual topology
Appendix Polars and the M ackey-A rens theorem
Notes
Problems

124
131
133
141
145
150
153
162
167
169
173

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III:


BOUNDED OPERATORS
/ . Topologies on bounded operators
2. Adjoints
3. The spectrum
4. Positive operators and the polar decomposition
5. Compact operators
6. The trace class and H ilbert-Schm idt ideáis
Notes
Problems

Vil:

THE SPECTRAL THEOREM
1. The continuous functional calculus
2. The spectral measures
3. Spectral projections
4. Ergodic theory revisited: Koopmanism
Notes
Problems

VIII:

182
185
188
195
198
206
213
216

221
224
234
237
243
245

UNBOUNDED OPERATORS
1. Domains, graphs, adjoints, and spectrum
2. Symmetric and self-adjoint operators: the basic criterion
fo r self-adjointness

3. The spectral theorem
4. Stone’s theorem
5. Formal manipulation is a touchy business:Nelson’s
example
6. Quadratic form s
7. Convergence o f unbounded operators
8. The Trotter product form ula
9. The polar decomposition fo r closed operators
10. Tensor producís
11. Three mathematical problems in quantum mechanics
Notes
Problems

249
255
259
264
270
276
283
295
297
298
302
305
312

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VI:

THE FOURIER TRANSFORM
318
326
332
338
339

SUPPLEMENTARY MATERIAL
11.2. Applications o f the Riesz lemma
III.1. Basic properties o f L» spaces
IV .3. P roof o f Tychonoff s theorem
IV A . The R iesz-M a rko v theorem fo r X = [0 , 1 ]
IV .5. Minimization o f functionals
V.5. Proofs o f some theorems in nonlinear functional analysis
VI.5. Applications o f compact operators
V III.7. Monotone convergence fo r form s
V III.8. More on the Trotter product form ula
Uses o f the máximum principie
Notes
Problems

List o f Symbols
Index

344
348
351
353

354
363
368
372
377
382
385
387

393
395

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1. The Fourier transform on 5 f(U n) and 6^'(Un),convolutions
2. The range o f the Fourier transform: Classicalspaces
3. The range o f the Fourier transform: Analyticity
Notes
Problems


Volume II:
IX
X

The F o u rie r T ransform
S e lf-A d jo in tn e ss a n d th e E xisten ce o f D ynam ics

Volume III:
XI

X III

Scattering Theory

S c a tte rin g T h eo ry

Volume IV:
X II

Fourler A nalysis, Self-Adjointness

A n alysis of Operators

P e rtu rb a tio n o f P o in t S p ectra
S p e c tra l A n a ly sis

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Contents of Other Volumes


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The beginner. .. should not be discouraged if. .. he finds that he does not have the prerequisites
for reading the prerequisites.
P. Halmos

1.1 Sets and functions

We assum e th a t the reader is fam iliar with sets and functions but it is
ap p rop riate to standardize our term inology and to introduce here abbreviations th a t will occur thro u g h o u t the book.
I f X is a set, x e X m eans th a t x is an element o f X ; x $ X m eans that x is
not in X . T he clause “ for all x in X " is abbreviated (V xe X ) and “ there
exists an x g X such t h a t ” is abbreviated (B x e JÍ). The symbol {jc|P(x)}
stands fo r the set o f x obeying the condition (or conditions) P(x). If A is a
subset o f X (denoted A cz X ), the sym bol X \A represents the com plem ent of
A in X , th a t is X \A = { x e X \ x $ A ). M ore generally, if A and B are subsets
o f X , then A \B = { x \ x e A , x $ B}. W hen we discuss sets with a topology, Á
will always denote the closure o f the set A . Finally, the set o f ordered pairs
K*> y ) Ix 6 X , y e 7} is called the C artesian product o f X and Y and is
denoted X x Y.
We will use the w ords “ fu n c tio n ” and “ m a p p in g ” interchangeably. In
order to em phasize th at certain functions / depend on two variables, we will
som etim es w rite /( • ,• ) • T he sym bol f ( - , y ) denotes the function of one
variable obtained by picking a fixed valué o f y for the second variable. A
1

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I: Preliminaries


set y, denoted b y / : X -» Y or x J + Y or x i-» /(x ) will have o n e a n d only one
valué in Y for each x e X . If A <= X , then f[ A ) = {/ ( x ) | x e A} is a subset of
7 and f ~ x[B] = ( x |/ ( x ) e B} is a subset o f X if B cz Y. f [ X ] will usually be
called the range o f f and will be denoted R an f X is called the domain o f / .
A function f will be called injective (or one-one) if for each y e R an f there
is at m ost x e X such th a t f ( x ) ~ y \ f is called surjective (or onto) if

Ran / = y. If / is both injective and surjective, we will say it is bijective. The
restriction o f / to a subset A o f its dom ain will be denoted by f \ A .
If X 3 A we define the characteristic function %A(x) as
x6A
x $A
There are two set theoretic notions which are slightly deeper than m ere
notation, so we will discuss them to some extent. A relation i ü o n a set X is a
subset R o f X x X ; if <x, y > e R, we say th at x isrelated (or R-related) to
y and write xR y.
D e fin itio n
(i)
(ii)
(iii)

A relation R is called an equivalence relation if it satisfies:

(Vx e X ) x R x [reflexive]
(Vx, y e X ) x R y implies y R x [symmetric]
(Vx, y, z e X ) x R y and y R z implies x R z [transitive]

The set o f elements in X th at are related to a given x e X is called the
equivalence class o f x, denoted usually as [x].
It is easy to pro ve:
Theorem 1.1
Let R be an equivalence relation on a set X . Then each
x e X belongs to a unique equivalence class.
Thus, under an equivalence relation, a set divides up in a natural way into
disjoint subsets.
Exam p le 1 (the integers m od 3)
Let X be the integers and write x R y

if x —y is a m últiple o f 3. This equivalence relation divides the integers into
three equivalence classes:
[0] —
- 6 , - 3 , 0 , 3, 6 ,...}
[1] = { ..., - 5 , - 2 , 1 , 4 , 7 , . . . }
[2] =
- 4 , - 1 , 2 , 5, 8 ,...}

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linear function will also be called an operator o r a linear transform ation. O ur
functions will always be single valued; so a function from a set X to another


Ex a m p ie 2 (the real projective line)
Let R denote the real line and Iet
X be the nonzero vectors in IR2 ( = IR x IR). We write x R y if there is some
a e IR with x = ay. The equivalence classes are lines through the origin (with
<0, 0 ) removed).

D e f in itio n
A relation on a set X which is reflexive, transitive, and antisym m etric (th at is, x R y and y R x implies x = y ) is called a partial ordering.
If i? is a partial ordering, we often w rite x < y instead o f xR y.
E x a m p le 3
Let X be the collection o f all subsets o f a set Y. Define
A -< B if A c: B. Then -< is a partial ordering.
We use the w ord “ partial ” in the above definition because two elements
of X need not obey x «< y o r y -< x. If fo r all x and y in X , either x -< or
y -< Jt, X is said to be linearly ordered. F o r example, R with its usual order <
is linearly ordered.

N ow suppose X is partially ordered by -< and Y c X . An element p e X
is called upper bound for 7 i f y < p for all y e Y. If m e X and m < .x implies
x = m , we say m is a maximal element o f X .
D epending on one’s starting point, Z o rn ’s lemma is either a basic assumption o f set theory or else derived from the basic assum ptions (it is equivalent
to the axiom o f choice). W e take Z o rn ’s lemma and the rest o f set theory as
given.
T h eo rem 1.2 (Z orn’s lemma)
Let X be a nonem pty partially ordered set
with the property th at every linearly ordered subset has an upper bound in X .
Then each linearly ordered set has som e upper bound that is also a maximal
elem ent o f X .
Finally, we will use H alm os’ | to indicate the conclusión o f a proof.

1.2 Metric and normed linear spaces

T hroughout this w ork, we will be dealing with sets of functions o r operators
or other objects and we will often need a way o f m easuring the distance

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Next, we discuss Z o rn ’s lemma.


between the objects in the sets. It is reasonable to define a notion o f distance
that has the most im portant properties o f ordinary distance in IR3.

(i)
(ii)
(iii)
(iv)

d (x, y)
d(x, y )
d(x, y)
d(x, z)

>
=
=
<

0
0 if and only if x = y
d(y, x)
d (x, y ) + d (y, z) [triangle inequality]

The function d is cafied a m etric on M.
We often cali the elements o f a m etric space points. N otice that a m etric
space is a set M together with a m etric function d\ in general, a given set X
can be m ade into a m etric space in different ways by em ploying different
metric functions. W hen it is not clear from the context which m etric we are
talking about, we will denote the m etric space by is explicitly displayed.

E x a m p le 1
Let M — IR" with the distance between two points x =
<xl s . . . , x„> and y = given by
d(x, y ) = J ( x t - y x)2 + • • • + (xH- y„)2

E x a m p le 2
Let M be the unit circle in IR2, th a t is, the set o f all pairs o f
real num bers < /,««, j8>, <«', / ¡ '» = V(A nother possible m etric is d 2[p, p '] — are length between the points p , p'
(see Figure 1.1).
pL.

F ig u re 1.1

The metrics dx and d2 .

^2

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D e fin itio n
A m etric space is a set M and a real-valued function d (*, •)
on M x M which satisfies:


Exam p le 3
Let M = C[0, 1], the continuous real-valued functions on
[0, 1] with either o f the m etrics
d i i f , g) = m ax | / O ) - g(x) |
*€[0,1]

d 2(f, g) = P | f ( x ) - g(x) j dx

Definition

A sequence o f elem ents {x:„}“=1 o f a m etric space said to converge to an element x e M , if í/(jc, jc„) -» 0 as « -> oo. We will often
denote this by x„— í_» x or lim ,,..^ x„ = x. If x n does not converge to x, we
will write x n—/ —>x.
In Exam ple 2, d x(p, p') < d 2(p, p') < n d x(p, p') which we will write
d x < d 2 < n d \. Thus p n . dt >p if and only if
í2
But in Exam ple 3, the
m etrics induce distinct notions o f convergence. Since d2 < d x
»/im p lie s
/ „ —íl-» /, but the converse is false. A counterexam ple is given by the functions
gn defined in Figure 1.2, which converge to the zero function in the m etric d2

F ig u re 1.2

The graph o f g„(x).

but which do not converge in the m etric d x. This may be seen by introducing
the im portant notion o f Cauchy sequence.
D e f in itio n
A sequence o f elem ents {*„} o f a m etric space called a Cauchy sequence if (Ve > 0)(3iV) n, m > N implies d(xn , Jtm) < e.
P ro p o siiio n

Any convergent sequence is Cauchy.

P roof G iven x„^> x and e, find N so n > N implies d(x„ , x) < e/2. Then
n, m > N implies d(x„, x:m) < d(x„ , x) + d(x, x m) < je + }e. |

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Now th a t we have a notion o f distance, we can say w hat we m ean by
convergence.


D e fin itio n
A m etric space in which all Cauchy sequences converge is
called complete.
F or example, IR is com plete, but O is not. It can be shown (Sections 1.3 and
1.5) that and R suggests w hat we need to do to an incom plete space X to m ake it
complete. W e need to enlarge X by adding “ all possible limits o f Cauchy
sequences.” The original space X should be dense in the larger space X
where:

D e fin itio n
A set B in a m etric space M is called dense if every m e M is a
ümit o f elements in B.
O f course, if the incom plete space is not already contained in a larger
complete space (like Q is contained in R) it is not clear w hat “ all possible
lim its” means. T h at this “ co m p letio n ” can be done is the contení o f a
theorem that we shall shortly State; but first some definitions:

D efiniüon
A function / from a m etric space <X, d > to a m etric space
< Y, p> is called continuous a t x if /(* „ ) <r,p>-» f { x ) whenever x„
> x.
We have already had an example o f a sequence o f elements in C[0, 1] with
f n.J 2— > 0 but f„ dl/ ~» 0. T hus the identity function from

to

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We now return to the functions in Figure 1.2. It is easy to see th at if n ^ m,
d\{g„, g m) = 1• T h u sg n is not a Cauchy sequence in not a convergent sequence. Thus, the sequence {g „} converges in but not in A lthough every convergent sequence is a Cauchy sequence, the following
example shows th at the converse need not be true. Let O be the rational
num bers with the usual m etric (that is, d(x, y ) — | x -- j>|) and let x* be any
irrational num ber (that is, x* e R\Q ). F ind a sequence of rationals x n with
x n -> x* in R. Then x„ is a Cauchy sequence o f num bers in Q , but it cannot
converge in Q to some y e Q (for, if x n - * y in Q, then x n - * y in IR, so we
would have y — x*).


D efínition
A bijection h from <X , d > to < Y, p> which preserves the
metric, th at is,
p(h(x), h(y)) = d(x, y)

Isom etric spaces are essentially identical as m etric spaces; a theorem concerning only the m etric structure o f ( X , d ) will hold in all spaces isom etric
to it.
We now state preciseiy in which sense an incom plete space can be fattened
out to be com plete:
T h eo rem 1.3
If find a com plete m etric space M so th a t M is isom etric to a dense subset of M .

Sketch o f p ro o f C onsider the C auchy sequences {x„} of elements o f M . Cali
two sequences, {x„}, {ym}, equivalent if lim ,,.^ d{x„ , y n) = 0. Let A? be the
family o f equivaíence classes o f Cauchy sequences under this equivalence
relation. One can show th at for any tw o C auchy sequences lim ,,.^ d(xn, y n)
exists and depends only on the equivalence classes o f {x„} and {y„}. This limit
defines a m etric on A? and A? is com plete. Finally, m ap M into M by taking x
into the constant sequence in which each x„ equals x. M is dense in M and
the m ap is isom etric. |
To com plete our discussion o f m etric spaces, we want to introduce the
notions o f open and closed sets. The reader should keep the exam ple of open
and closed sets on the real line in m ind.
D efínition

Let <X, d > be a m etric space:

(a) The set { x |x e X , d(x, y) < r} is called the open ball, B ( y ; r), o f radius
r about the p oint y.
(b) A set O cz X is called open if (Vy e 0 )(3 r > 0) B ( y ; r) cz O.
(c) A set N c: X is called a neighborhood o f y e N if B (y ; r) c N for some
r > 0.
(d) Let E c Al . A point x is called a limit point o f E, if (V r > 0)
B (x; r) n (£\{x}) 7 ^ 0 , th a t is, x is a limit p oint o f E if E contains points
other than x arbitrarily near x.
(e) A set F <= X is called closed if F contains all its limit points.
( f ) If G cz X , x e G is called an interior point o f G, if G is a neighborhood
o f x.

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is called an isometry. It is autom atically continuous.

said to be isometric if such an isom etry exists.


The reader can prove for him self the following collection o f elem entary
statem ents:
Let
(a) A set, O, is open if and on!y if X \ 0 is closed.
(b) x m——■> x if and only if fo r each neighborhood N o f x, there exists an
M so th at m > M implies x m e N.
(c) The set o f interior points o f a set is open.
(d) The unión o f a set E with its limit p o in ts is a closed set (denoted by E
and called the closure of E).
(e) A set is open if and only if it is a neighborhood o f each o f its points.
One o f the m ain uses o f open sets is to check for convergence using
Theorem I.4.b and in particular to check for continuity via the following
criteria, the p ro o f o f which we lea ve as an exercise:
Th eorem 1.5
A function /(•) from a m etric space X to another space Y
is continuous if and only if for all open sets O a Y , / -1 [O] is open.
Finaliy, we w arn the reader th at often in incom plete m etric spaces, closed
sets may not appear to be closed at first glance. F or example, [^, 1) is closed in
(0, l) (with the usual metric).
We complete this section with a discussion o f tw o o f the central concepts of
functional analysis: norm ed linear spaces and bounded linear transform ations.
D e fm itio n
A normed linear space is a vector space, V, over IR (or C)
and a function, ||-|j from V to IR which satisfies:
(i)
(ii)

(iii)
(iv)

í|u¡| > 0 for all v in V
\\v\\ — 0 if and only if v — 0
liai'H = ¡a | ||v || for al! v in V and a in IR (or C)
||u + w |¡ < ¡|y|| + IM¡ for all u and win V

D e fm itio n
A bounded-linear transform ation (o r bounded operator) from
a norm ed linear space <K,, || ||t > to a norm ed linear space <F2 , || ¡|2> is a
function, T, from V¡ to V2 which satisfies:
(i)
(ii)

T(otv + pw) = a T(v) + fiT(w ) (Vu, w e F)(Va, fi e IR or C)
F o r some C > 0, ||T y ||2 < C||t>|li

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Th eo rem 1.4


The sm allest such C is called the norm o f T, w ritten ||T || or ||T ||1>2. Thus
\\T\\ =

sup |!Tü|¡2
l|w||| = i

Since we will study these concepts in detail later, we will n ot give many

examples now b u t merely note th at IR" with the norm
||<x1, . . . , * fl>|| = V l * i l 2 + ’ - ’ + \x n\2

ll/ilco=

sup

|/ ( x ) |

*6[0, 1]

or

ll/Ü! =

f

\f(x )\d x

•'o

are norm ed linear spaces. Observe also th a t any norm ed linear space <F, ¡| • |¡>
is a m etric space when given the distance function d(v, w) — ||i? — w\\. There
is thus a notion o f continuity o f functions, and for linear functions this is
precisely captured by bounded linear transform ations. The p ro o f o f this fact
is left to the reader.
T h e o re m 1.6
Let T be a linear tra n sfo rm a ro n between two norm ed
linear spaces. The following are equivalent:
(a)

(b)
(c)

T is continuous a t one point.
T is continuous a t all points.
T is bounded.

D efinition
W e say space in the induced m etric.
If <X, || • ||> is a norm ed linear space, then X has a com pletion as a metric
space by Theorem 1.3. U sing the fact th a t X is dense in X , it is easy to see that
X can be m ade into a norm ed linear space in exactly one natural way.
All these concepts are well illustrated by the following im p o rtan t theorem
and its p r o o f :

T h eo rem 1.7 (the B.L.T. theorem )
Suppose T is a bounded linear trans­
fo rm a ro n from a norm ed linear space <Vj, || • jli > t o a com plete norm ed linear
space <F2 , || *||2>- T hen T can be uniquely extended to a bounded linear
transform ation (with the same bound), T, from the com pletion of Vl to
<y2 , ih i2>.

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and C[0, 1] with either the norm