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INTRODUCTORY

FUNCTIONAL

ANALYSIS

WITH

~

APPLICATIONS

Erwin Kreyszig

University of Windsor

JOHN

WILEY

&

SONS

New York Santa Barbara London Sydney

Toronto

Copyright © 1978, by

John

Wiley & Sons. Inc.

All rights reserved. Published simultaneously in Canada.

No

part

of this book may be reproduced by any means,

nor transmitted,

nor

translated into a machine language

without the written permission of the publisher.

Library

of

Congress Cataloging in Publication Data:

Kreyszig, Erwin.

Introductory functional analysis with applications.

Bibliography: p.

1.

Functional analysis. I. Title.

QA320.K74

515'.7 77-2560

ISBN 0-471-50731-8

Printcd in thc Unitcd States of America

10

9 H 7 6 5 4

~

2 I

PREFACE

Purpose of the book. Functional analysis plays an increasing role in

the applied sciences as well as in mathematics itself. Consequently, it

becomes more and more desirable to introduce the student to the field

at

an early stage of study. This book

is

intended to familiarize the

reader with the basic concepts, principles and methods of functional

analysis and its applications.

Since a textbook should be written for the student, I have sought

to bring basic parts of the field and related practical problems within

the comfortable grasp of senior undergraduate students

or

beginning

graduate students of mathematics and physics. I hope that graduate

engineering students may also profit from the presentation.

Prerequisites. The book

is

elementary. A background in under-

graduate mathematics, in particular, linear algebra and ordinary cal-

culus,

is

sufficient as a prerequisite. Measure theory

is

neither assumed

nor

discussed. No knowledge in topology

is

required; the few consider-

ations involving compactness are self-contained. Complex analysis

is

not

needed, except in one of the later sections (Sec. 7.5), which

is

optional, so that it can easily be omitted. Further help

is

given in

Appendix 1, which contains simple material for review and reference.

The

book should therefore be accessible to a wide spectrum of

students and may also facilitate the transition between linear algebra

and advanced functional analysis.

Courses. The book

is

suitable for a one-semester course meeting five

hours

per

week

or

for a two-semester course meeting three hours

per

week.

The book can also be utilized for shorter courses. In fact, chapters

can be omitted without destroying the continuity

or

making the rest of

the book a torso (for details see below).

For

instance:

Chapters 1 to 4

or

5 makes a very short course.

Chapters 1 to 4 and 7

is

a course that includes spectral theory and

other

topics.

Content and arrangement. Figure 1 shows that the material has

been

organized into

five

major blocks.

III

I'r('j'(/('('

----

~----.

SPUCIIS

""d

Oponttors

Chaps. 1

to

3

Metric spaces

Normed and Banach spaces

'I

Linear operators

I

I nner product and Hilbert spaces

i

!

I

Fundamental Theorems

Chap. 4

Hahn-Banach theorem

Uniform boundedness theorem

Open mapping theorem

Closed graph theorem

I

!

I

Further Applications

Chaps. 5

to

6

I

Applications

of

contractions

J

I

Approximation theory

j

I

t

t

Spectral Theory

Chaps, 7

to

9

Basic concepts

Operators

on

normed spaces

,

I

Compact operators

I

Self~adjoint

operators

)

!

Unbounded Operators

Chaps. 10

to

11

Unbounded operators

Quantum mechanics

Fig. 1. Content and arrangement of material

Hilbert space theory (Chap.

3)

precedes the basic theorems on

normed and Banach spaces (Chap.

4)

because it

is

simpler, contributes

additional examples

in

Chap. 4 and, more important, gives the student

a

better

feeling for the difficulties encountered

in

the transition from

Hilbert spaces to general Banach spaces.

Chapters 5 and 6 can be omitted. Hence after Chap. 4 one can

proceed directly to the remaining chapters

(7

to 11).

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