Functional analysis for the applied sciences

439 Pages·2020·1.792 MB·English

by  Morosanu G

Universitext
Gheorghe Moroşanu
Functional
Analysis for
the Applied
Sciences
Universitext
Universitext
Series editors
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
CarlesCasacuberta
UniversitatdeBarcelona,Barcelona,Spain
JohnGreenlees
UniversityofWarwick,Coventry,UK
AngusMacIntyre
QueenMaryUniversityofLondon,London,UK
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
ClaudeSabbah
E´colePolytechnique,CNRS,Universite´ Paris-Saclay,Palaiseau,France
EndreSu¨li
UniversityofOxford,Oxford,UK
WojborA.Woyczyn´ski
CaseWesternReserveUniversity,Cleveland,OH,USA
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Gheorghe Moros¸anu
Functional Analysis
for the Applied Sciences
GheorgheMoros¸anu
RomanianAcademyofSciences
Bucharest,Romania
DepartmentofMathematics
Babes-BolyaiUniversity
Cluj-Napoca,Romania
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-030-27152-7 ISBN978-3-030-27153-4 (eBook)
https://doi.org/10.1007/978-3-030-27153-4
MathematicsSubjectClassification(2010):32A70
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Dedicated to my wife, Carmen
Preface
The goal of this book is to present in a friendly manner some of the
main results and techniques in Functional Analysis and use them to
explore various areas in mathematics and its applications. Special
attention is paid to creating appropriate frameworks towards solving
different problems in the field of differential and integral equations. In
fact, the flavor of this book is given by the fine interplay between the
tools offered by Functional Analysis and some specific problems which
are of interest in the Applied Sciences.
The table of contents of the book (see below) offers a fairly good
description of the material. In contrast with other books in the field,
we present in Chap.1 the real number system, describing the Cantor–
M´eraymodelwhichismostappropriateforourpurposeshere. Indeed,
it is based on a completion procedure, allowing the extension from ra-
tional numbers to real numbers. This procedure involves the concepts
of limit and infinity that are specific to analysis. We consider the
Cantor–M´eray construction as the corner stone of mathematical anal-
ysis, which is why we pay attention to this subject which is usually
assumed well known.
In order to help the reader to understand the richness of ideas and
methods offered by Functional Analysis, we have included a section of
exercises at the end of each chapter. Some of these exercises supple-
ment the theoretical material discussed in the corresponding chapter,
while others are mathematical problems that are related to the real
world. Some of the exercises are borrowed from other books, being
reformulated and/or presented in a form adapted to the needs of the
correspondingchapter. Wedonotindicatethebookswhereindividual
exercises come from, but all those sources are included into the refer-
ence list of our book. In any event, we do not claim originality in such
cases. Other exercises were invented by us to offer the reader enough
vii
viii Preface
material to understand the theoretical part of the book and gain ex-
pertise in solving practical problems. In the last chapter of the book
(Chap.12), we provide solutions to almost all exercises. This is incon-
trast to many other books which include exercises without solutions.
For easy exercises, we provide hints or final solutions, and answers to
very easy exercises are left to the reader. I encourage everybody to
spend some time working on an exercise before looking at its solution.
We shall refer to an exercise by indicating the chapter and exercise
numbers (and not the section number). For example, Exercise 11.3
will mean Exercise 3 in the last section of Chap.11 (which is Sect.11.3
in this case).
The book is addressed to graduate students and researchers in
applied mathematics and neighboring fields of science.
I would like to thank the anonymous reviewers whose pertinent
comments improved the initial version of the book.
Special thanks are due to a former American student of mine, Ivan
Andrus, who wrote the first draft of the present book as lecture notes
for my Functional Analysis lectures in 2010. He also carefully checked
the final version of the book and suggested several minor changes.
I am also indebted to my former student Liviu Nicolaescu for read-
ing the first part of the book and correcting some errors.
Last but not least, I would like to thank Mrs. Elizabeth Loew,
Executive Editor at Springer, for our very kind cooperation that led
to the successful completion of this book project.
Cluj-Napoca, Romania Gheorghe Moro¸sanu
Contents
1 Introduction 1
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . 15
1.5 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Metric Spaces 31
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Continuous Functions on Compact Sets . . . . . . . . . 44
2.5 The Banach Contraction Principle . . . . . . . . . . . 55
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 The Lebesgue Integral and Lp Spaces 65
3.1 Measurable Sets in Rk . . . . . . . . . . . . . . . . . . 65
3.2 Measurable Functions . . . . . . . . . . . . . . . . . . . 71
3.3 The Lebesgue Integral . . . . . . . . . . . . . . . . . . 75
3.4 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Continuous Linear Operators and Functionals 89
4.1 Definitions, Examples, Operator Norm . . . . . . . . . 89
4.2 Main Principles of Functional Analysis . . . . . . . . . 93
4.3 Compact Linear Operators . . . . . . . . . . . . . . . . 96
4.4 Linear Functionals, Dual Spaces, Weak Topologies . . 97
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 104
ix
x Contents
5 Distributions, Sobolev Spaces 107
5.1 Test Functions . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Friedrichs’ Mollification. . . . . . . . . . . . . . . . . . 112
5.3 Scalar Distributions . . . . . . . . . . . . . . . . . . . . 119
5.3.1 Some Operations with Distributions . . . . . . 121
5.3.2 Convergence in Distributions . . . . . . . . . . 122
5.3.3 Differentiation of Distributions . . . . . . . . . 125
5.3.4 Differential Equations for Distributions . . . . . 131
5.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 143
5.5 Bochner’s Integral . . . . . . . . . . . . . . . . . . . . . 149
5.6 Vector Distributions, Wm,p(a,b; X) Spaces . . . . . . . 155
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Hilbert Spaces 165
6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.2 Jordan–von Neumann Characterization Theorem . . . 168
6.3 Projections in Hilbert Spaces . . . . . . . . . . . . . . 171
6.4 The Riesz Representation Theorem . . . . . . . . . . . 175
6.5 Lax–Milgram Theorem . . . . . . . . . . . . . . . . . . 180
6.6 Fourier Series Expansions . . . . . . . . . . . . . . . . 186
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7 Adjoint, Symmetric, and Self-adjoint Linear
Operators 201
7.1 The Adjoint of a Linear Operator . . . . . . . . . . . . 201
7.2 Adjoints of Operators on Hilbert Spaces . . . . . . . . 204
7.2.1 The Case of Compact Operators . . . . . . . . 205
7.3 Symmetric Operators and Self-adjoint Operators . . . 209
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8 Eigenvalues and Eigenvectors 217
8.1 Definition and Examples . . . . . . . . . . . . . . . . . 217
8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . 219
8.3 Eigenvalues of −Δ Under the Dirichlet Boundary
Condition . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.4 Eigenvalues of −Δ Under the Robin Boundary
Condition . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.5 Eigenvalues of −Δ Under the Neumann Boundary
Condition . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.6 Some Comments . . . . . . . . . . . . . . . . . . . . . 232
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 239