Functional Analysis in Applied Mathematics and Engineering

311 Pages·2018·9.2 MB·English

by  Pedersen

Functional Analysis
in Applied
Mathematics
and Engineering
Studies in Advanced Mathematics
Series Editor
STEVEN G. KRANTZ
Washington University in St. Louis
Editorial Board
R. Michael Beals
Gerald B. Folland
Rutgers University
University of Washington
Dennis de Turck
William Helton
University of Pennsylvania
University of California at San Diego
Ronald DeVore
Norberto Salinas
University of South Carolina
University of Kansas
iMwrence C. Evans
Michael E. Taylor
University of California at Berkeley
University of North Carolina
Titles Included in the Series
Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping
John J. Benedetto, Harmonic Analysis and Applications
John J. Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications
Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex
Goong Chen andJianxin Zhou, Vibration and Damping in Distributed Systems, Voi. 1:
Analysis, Estimation, Attenuation, and Design. Voi. 2: WKB and Wave Methods,
Visualization, and Experimentation
Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic
Functions
John P. DAngelo, Several Complex Variables and the Geometry of Real Hypersurfaces
Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of
Functions
Gerald B. Folland, A Course in Abstract Harmonic Analysis
José García-Cuerva, Eugenio Hernández, Fernando Soria, and J osé-Luis Torrea, Fourier
Analysis and Partial Differential Equations
Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index
Theorem, 2nd Edition
Alfred Gray, Modem Differential Geometry of Curves and Surfaces with Mathematica, 2nd
Edition
Eugenio Hernández and Guido Weiss, A First Course on Wavelets
Steven G. Krantz, Partial Differential Equations and Complex Analysis
Steven G. Krantz, Real Analysis and Foundations
Kenneth L Kuttler, Modem Analysis
Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition
John Ryan, Clifford Algebras in Analysis and Related Topics
Xavier Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential
Operators
Robert Strichartz, A Guide to Distribution Theory and Fourier Transforms
André Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones
James S. Walker, Fast Fourier Transforms, 2nd Edition
James S. Walker, Primer on Wavelets and their Scientific Applications
Gilbert G. Walter, Wavelets and Other Orthogonal Systems with Applications
Kehe Zhu, An Introduction to Operator Algebras
Functional Analysis
in Applied
Mathematics
and Engineering
MICHAEL PEDERSEN
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data
Pedersen, Michael.
Functional analysis in applied mathematics and engineering /
Michael Pedersen.
p. cm.—(Studies in advanced mathematics)
Includes bibliographical references and index.
ISBN 0-8493-7169-4 (alk. paper)
1. Functional analysis. I. Title II. Series.
QA320.P394 1999
515'.7-^c21 99-37641
CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
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International Standard Book Number 0-8493-7169-4
Library of Congress Card Number 99-37641
Printed in the United States of America 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
A cknowledgments
I would like to thank my colleagues, Prof.O.j0rsboe and Prof.L.MejIbro,
for correcting a number of errors and for collecting so many good exercises.
Also, a warm thanks to my former students. Dr.Roger Krishnaswamy, and
Dr.Michael Danielsen for the material on the Hilbert Uniqueness Method,
and Dr.Anda Binzer for assistance.
Michael Pedersen

Preface
Functional analysis is a ’’supermodel” for mathematical analysis that has
been developed in particular after the Second World War. The general idea
is to construct an abstract framework suited to deal with various problems
from mathematical analysis that perhaps at first glance seem to have noth­
ing in common, but when all ’’inessentials” are stripped away, appear to be
similar. The language from linear algebra is used to describe the setting of
the problems, and this notational simplification is both the power and dan­
ger of functional analysis: many formulations and formulas look extremely
simple but typically reflect highly delicate subjects.
The present book. Functional Analysis in Applied Mathematics and Engi­
neering j is primarily written to cover the course in functional analysis given
at The Technical University of Denmark. This is a one-semester (14 weeks)
course, and in order to pass, the students will have to do 12 obligatory sets
of exercises and have 10 of them judged ’’passed”. Therefore, there is a
large number of exercises, many of them are folklore in the sense that they
apparently can be found in all basic functional analysis courses. Solutions
to all the exercises can be obtained by contacting the author by e-mail. The
course is recommended for students on third year or later having a solid
background in basic mathematical analysis or corresponding mathematical
“maturity”.
The aim of the book is to get as fast as possible to the operators on
Hilbert spaces with the spectral theorem as the final highlight, without
compromising on the mathematical rigor. The course will provide a solid
background for the understanding of the problems encountered in applied
mathematics and engineering, focusing in particular on the abstract for­
mulations of partial differential equations in a Hilbert space setting. A
lot of sophisticated theory that should be incorporated in a pure math,
functional analysis course is left out, such as measure theory and more ad­
vanced topological considerations. Instead, we typically argue by a density
or completion statement. We consider only separable Hilbert spaces since
nonseparable Hilbert spaces occur very rarely in applications.
Chapters 1-6 are mandatory; they provide the theory that is necessary
in order to read and understand texts and papers in modern applied math­
ematics. The basic idea of these chapters is to ’’teach” the subject, and we
emphasize that in order to fully understand the theory, it is vital to cal­
culate a large number of the exercises. The exercises are ordered roughly
following the chapters, some of them are easy applications of the theory,
others are more interesting and it is highly recommended to do at least
50-60 of them. Therefore, it is also recommended that some of the time of
the regular lectures is used to discuss the aspects of the theory developed
in the exercises. The last chapters are introductions to different subjects
from functional analysis and can be read independently. The main purpose
of these chapters is to serve as an appetizer to a more specialized study
of functional analysis and control theory for partial differential equations.
There are no exercises to these chapters, and since the purpose is now
completely different from the first part of the book and the theory rather
advanced, a number of proofs are omitted and references are much more
common. In Chapters 12 and 13, however, detailed proofs of recent results
that so far only are published in scientific journals are incorporated. The
idea is to take one or two subjects from these chapters as examples of ad­
vanced applications as the conclusion of the regular 14-week course, but it
is also possible to use the material in a more advanced applied functional
analysis course.
There is a vast literature on functional analysis, the main part covering
what could be called general functional analysis in the sense that the aim
is to state and prove the theory in maximal generality, and the reader is
typically required to have a good knowledge of measure theory and gen­
eral topology. This kind of littérature is well suited for pure mathematics
students, where the requirements are essential also for many other subjects
during the period of study. For an engineering student who must be ed­
ucated in a number of other technical disciplines, it will typically not be
possible to meet the requirements of general functional analysis at the time
in the study where the mathematical problems encountered in the techni­
cal diciplines become so complicated that they actually require functional
analytic considerations. With the present book, we try to meet the back­
ground of typical engineering or applied mathematics students and give a
short but firm introduction to the subject. As mentioned above, we do not
compromise on the mathematical rigor and the ’’construction” of the book
is classical, with theorems and detailed proofs. The main part of the book
is focused on operators on Hilbert spaces, and closed unbounded operators
are introduced to provide the proper setting for the differential equations
from applied mathematics and engineering, some of which are presented in
the last part of the book.
Contents
1 Topological and Metric Spaces 1
1.1 Some Topology 1
1.2 Metric Spaces 2
2 Banach Spaces 11
201 Normed Vector Spaces 11
202 £P-spaces 0 0 0 0 0 0 0 12
203 Infinite Dimensional Spaces 19
3 Bounded Operators 23
301 Basic Properties 0 0 0 0 0 0 23
302 Bounded Linear Operators 0 26
4 Hilbert Spaces. 33
401 Inner Product Spaces 0 0 0 0 0 0 33
402 Hilbert Spaces 0 0 0 0 0 0 0 0 0 36
403 Construction of Hilbert Spaces 42
4.4 Orthogonal Projection and Complement 49
4o5 Weak Convergence 0 0 0 0 0 0 0 0 0 0 0 0 52
5 Operators on Hilbert Spaces 55
501 The Adjoint of a Bounded Operator 55
502 Compactness and Compact Operators 61
503 Closed Operators 0 0 0 0 0 0 0 0 0 0 0 0 65
5.4 The Adjoint of an Unbounded Operator 73
6 Spectral Theory 77
601 The Spectrum and the Resolvent 78
602 Operator-Valued Functions o 0 0 86