Functional Analysis

623 Pages·2008·25.74 MB·English

by  Balmohan Vishnu Limaye

FUNCTIONAL ANALYSIS
Second Edition

Balmohan V. Limaye

Professor of Mathema
Indian Institute of Technology Bombay

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Preface

‘This book is intended to serve as a text for a variety of introduc.
tory courses in functional analysis, primarily at the Master's level
in a mathersatics programme. Functional analysis embodies the ab-
stract approach to analysis. It brings out the essence of a problem
by ce

of apparently unrelated topics. It highlights the interplay between
algebraic structures and distance structures. Since functional analy-
ss provides « major link between mathematics and

scientists and enlightened engincers may also find thi book useful

ing out unnecearary details and thus gives a unified treatment

sppliestions,

‘The book is based on lecture courses given by the author mainly st
the Indian Institute of Technology Bombay.

Prerequisites. The reader of this buok is expected to know set the

loretic concepts, elements of linear algebra and rudiments of meteic

spaces. These topics are reviewed in the first three sections. ‘The

fourth section contains @ brief treatment of the theory of Lebesgue

‘measure on the real line, While this treatment is not indispensable

for understanding the subject matter of the book, itis useful in illus-

trating various concepts, ‘The book is elementary in the aense that

neither general topological spaces-nor arbitrary measure spaces are
ireussed anywhere

Plan. The general plan of the book it to impose a distance structure
ona linear space, exploit it to the fullest and introduce alditional fea
tures only when one cannot get any further without them. Thus the
basic structure of « normed space is introduced in Section 5, contin
linear maps on it are

Hahn-Banach theorems are proved in Section 7
spaces (that

completeness of « norm is exploited to obtain four major theorems,
‘namely the uniform boundedness principle, the closed graph theorem,

.cussed in Section 6 and the important
omplete normed
Section 8 and the

Banach spaces) are introdueed
w Preface

the open mapping theorem and the bounded inverse theorem in Sec:
tions 9, 10 and 11. The consideration ofthe spectrum of a bounded
operator in Section 12 as well as the account of duals and transposes
in Section 18 often refer to Banach spaces. While the basie theory of
compact operators given in Section 17, 18 and 19 works well on any
norined space, the discussion on appreximate solutions in Section 20
does need the normed space to be complete, Finally, the geomet
cally significant structure of an inner product space is introduced in
Section 21 and complete inner product spaces (that i, Hilbert spaces)
we studied in Sections 22, 23 and 24, Bounded operators on Hilbert
spaces are considered in Sections 25 to 28 with special reference to
the adjoint operation.
Since some of the readers may be interested only in the richest
structure of « Hilbert space, the material in Sections 21 to 28 is kept
independent of Sections 5 to 20,

Courses. Depending on the number of hours available for instruction
and on the maturity ofthe audience, several courses ean be developed
by choosing appropriate sections in the book. Some suggestions

(3) A course on Banach spaces and operators on them based on
Sections 5 to 14 with additional material from Section 15 on weak
‘and weak* convergence, Section 16 on reflesivty, Appendix A on
fixed points, or Appendix B on extreme points
i) A couree on Hilbert spaces and operators on them based on
Sections 21, 2 and 24 to 28 with addtional material from Section 23
fon approximation and optimization, Appendix C on Sturm-Liowville
problems, or Appendix D on unbounded operators and quantum me-

chanics.

(ii) A course on Banach and Ililbert spaces based on Sections 5

to and 21 to.24 with additional maveral from Section 23 on approx-

mation and optimization, Appendix A on fixed points, or Appendix
1B on extreme points

‘ay & courte on bounded operators and compact operators on
Preface ‘

normed spaces bused on Sections § to 18 and 17 to 20.
(*) A comprehensive courte on Banach and Hilbert spaces, and
bounded and compact operators on thein based on Sections 5 to 13,
17 to 19, 21, 22 and 24 to 28 with additional material from Sections
14, 15, 16, 2, 28 or Appendices A, B, C, D.
‘The essential interdependence of various sections and appeniices
is indicated in the fllowing diagram,

Seat tA
|
Secs fos _ Se alk 2
i ~1 TT) _— “7
seadio saais aia] selzs Sods
~ | Sees. 25 tw 27
seater | |
|
seattor | se |
Ape App ayn

Of course not everything ina given section need be covered. Judi
ious choice of material (like selecting only some of the corollaries and
fone or two applications of an important theurem) is very much called
for. Here is how a course should not be developed: Inchude all major
‘theorems in the book and exclude all examples and appli

Arrangement. ‘The book contains seven chapters and four appen-
dices. Bach chapter begins with a b
ite contents, placing them in context aud pointing out some novel
features. There are four sections in each chapter, A typical sect
dealing with a major result i organized as follows. It begint with
8 discussion of a basic feature of the main result. A technical re-
sult fllows in the form of «lemma, ‘The proof of the main result is
accomplished by using thie lemma in conjunction with other facts a

4 introduction summarising
4“ Preface

ready developed. Two types of examples follow next. One type shows
that some of the hypotheses ike completeness, finite dimensionality
ee, cannot be dropped. The other type shows how in particular easet
snteresting consequences are obtained when the hypotheses are, in
fact, satisfied. Both types of examples are an integral part of that
section and sometimes involve numerical calculations. Occasionally,
there is a subsection which forms a part of the section by itself and
sed if there is not enough time to cover it. ‘The lemma,
theorems, corollaries and examples are numbered for the purpose of
cnoaa eference, but the definitions are not. An extensive index at the
end uf the book can be used to lcate the definitions of new terins. A
Uist of aymbels which precedes the index may also be helpful. At the
ond ofeach section, a long list of problems, based sequentially on the
topirs covered in that section, is given. All problems are in the form
of statements to be established, obviating the need for a separate list
of answers. The problems range fom the most easy tothe very chal
longing, for which hints are often provided. Retults based on these
problems are not used later in the text of the book. Thus the reader
ie not required to solve these problems, although he is strongly urged
to attempt as many of them as he ean ia order to gain insight.

can be 0

‘The four appendices at the end of the book are of « diferent
natures compared to the sections. They point to further areas where
the beginnings madein the book ean lead. In particular, not all results
stated in these appendices are proved and no problems are listed

Approach, We have in general preferred a geometric approach to
far analytic one It has dictated the kind of proofs given for some
‘ems lke the Hahn-Banach theorems of Section T. A few
are included to help a reader

snag th

schematic figures, drawn using GL
suslige the relevant arguments. Also, the essentially applied approach
‘of constructing a solution, or atleast an approximate one, is adopted
rather than just proving the existence or the wniqueness of such a
solution, For example, see 124, Section 20; 24.2, 27.5(b), 287
Preface vi

Applications. Many results proved in the book are applied to di-
verse areas of mnathematis such as classical alysis (generalized lim
ite in 7.12, Fourier series in 9.4, 11.2, 15.5 and 22.8(b), convergence
of quadrature formulae ia 9.5, summabilty methods in 9.7, the mo-
‘ment problem of Hausdorff in 14.7, the Fourier Plancherel transform
in 26.6), differential and integral equations (the perturbation tech
rique in Section 10, Fredholm integeal equations in 19.3, 19.4 and
28.8(b), Sturm-Liowvlle problems in Appendix C), probability theory
(Helly’s selection principle in 15.7), approximation and optimization
theory (best approximation in Section 23), fixed point theory (Ap

pendix A), convex programming (Appendix B), ax well as to other
ranches of scence suchas optinal conta theory (quadratic lous
conta for dynamical ystems in 2246), signal analy (after 26.6)
tnd quantum rrchanics (Appendix D),

‘Treatment. Several standard books on functional analysis have been
consulted to treat specific topics covered in the book, Also, recent
research wotk is cited to indicate the present foontiers of the subject
ding op

matter, especially of some long n questions.

In order to give a historical perspective, statements of most major

results are preceded by the names uf mathematicians who discovered
them and the years of their discovery

Coverage. The book contains enough significant material fora fist
course in functional analysis, Any introductory book has to leave
cout some ofthe finest topes. ‘The present book is no exception. Por
example, the distribution theory, the spectral theory of bounded sel
adjoint operators or the theory of Banach algebras do not find a place
haere, These are all parts of what is knovn as linear functional analysig
‘The fast groming subject of nonlinear functional analysis (including
calculus in Banach spaces) is far beyond the scope of this book.

Second edition. This edition differs to some extent from the fi

cedition in style as well as an contents. Hh
vis Preface

able than the first. It gives more motivation and less formulae. The
sections in the first edition on the closed graph theorem, the open
mapping theorem, the spectrum of « bounded operator, the Fredholm
alternative and the integral equations are eorganiaed. While consid.
ring the spectrum of a hounded operator or of a compact operator,
the underlying normed space is not required to be complete. The
{following additions are made: construction of various quadrature for
_ulae in Section 9, relations between the zero spaces and the range
spaces of a bounded linear map and its transpose (13.7), the dosed
range theorem of Banach (18.10), an elementary proof of Eberlein

theorem on reflexivity (16.5), equivalence of boundedness and weak
boundedness of a sequence in a Hilbert space (24.8), a discussion of
Hilbert-Schmidt operators (28.2) and an entire section on apprea

‘mate solutions (Section 20). Some results and examples from the first
edition are relegated to the problems in the second edition, The lists
of problems are now considerably longer. In order to keep the book
sine manageable, the chapter on the spectral analysis of self-adjoint
operators is reluctantly dropped

Acknowledgments, The book was originally written in 1980 under
1 project sponsored by the University Grants Commission, New Deli
During the past fifteen years, several students and teachers have sent
sme their reactions to the frst edition. I am grateful to all of them.
‘The second edition was supported by the Curriculum Development
Programme of the Indian Institute of Technology Bombay. Some of
my friends, expecially M. Thamban Nair and P. Shunmugaraj, have
taken great pains to read the new version and suggest improvements
[Lam indeed indebted to them. I thank C. L. Anthony for processing
the manuscript using ATRX. Sabbatical leave granted by the 11. T
Bombay and encouragement given by my wife Nirmala were crucial
{or the completion of this edition.

August 8, 1995 Balmohan V. Limaye
Contents

Chapter 1

Chapter IT

(Chapter I

Chapter IV

Preliminaries 1
Relations on a Set 1

Linear Spaces and Linear Mape 5

Metric Spaces and Continuous Functions 19
Lebesgue Measure and Integration 42

Fundamentals of Normed Spaces 62

Normed Spaces 62
Continuity of Linear Maps 83
Haha-Banach Theorems 104
Banach Spaces 124

Bounded Linear Maps on Banach Spaces 138

9 Uniform Boundedness Principle 138
10 Closed Graph and Open Mapping Theorems 166
11 Bounded Inverse Theorem 182

12 Spectrum of a Bounded Operator 192

Spaces of Bounded Linear Functionals 216

18 Duals and Transposes 216
14. Duals of 1%([a,8) and C((a,8]) 235
15, Weak and Weak* Convergence 260
16 Reflexivity 280
Chapter V

Chapter VI

Chapter VIE

Appendix
Appendix
Appendix
Appendix

Bibliography 591

‘Compact Operators on Normed Spaces 302

w
18
10
20

302
Spectrum of a Compact Operator S17
Fredholm Alternative 333
Approximate Solutions 346

Compact Linear Mi

Geometry of Hilbert Spaces 367

a
2
2
m4

Inner Product Spaces 367
Orthonormal Sets 381

Approximation and Optimisation 402
Projection and Riess Representation Theorems 420

Bounded Operators on Hilbert Spaces 441

25

woes

Bounded Operators and Adjointe 442
Normal, Unitary and Self-Adjoint Operators 460
Spectrum and Numerical Range 483
Compact Self-Adjoint Operators 504

Fixed Pointe 528
Extreme Points 54

Strm-Liouville Problems 553

Unbounded Operators and Quantum Mechanics 571

List of Symbols 597

Index 601