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Functional Analysis in Applied Mathematics and Engineering PDF

311 Pages·2018·9.2 MB·English
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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 efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2000 by CRC Press LLC No claim to original U.S. Government works 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

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Presenting excellent material for a first course on functional analysis , Functional Analysis in Applied Mathematics and Engineering concentrates on material that will be useful to control engineers from the disciplines of electrical, mechanical, and aerospace engineering.This text/reference discuss
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