Theoretical Numerical Analysis: A Functional Analysis Framework

467 Pages·2001·1.811 MB·English

by  Kendall Atkinson, Weimin Han

9
Texts in Applied Mathematics
Editors
J.E. Marsden
L. Sirovich
M. Golubitsky
Advisors
G. Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton
Springer
NewYork
Berlin
Heidelburg
Barcelona
HongKong
London
Milan
Paris
Singapore
Tokyo
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Kendall Atkinson Weimin Han
Theoretical Numerical Analysis
A Functional Analysis Framework
With25Illustrations
1 3
KendallAtkinson WeiminHan
DepartmentofMathematics DepartmentofMathematics
DepartmentofComputerScience UniversityofIowa
UniversityofIowa IowaCity,IA52242
IowaCity,IA52242 USA
USA [email protected]
[email protected] and
DepartmentofMathematics
ZhejiangUniversity
Hangzhou
People’sRepublicofChina
SeriesEditors
J.E.Marsden L.Sirovich
ControlandDynamicalSystems,107-81 DivisionofAppliedMathematics
CaliforniaInstituteofTechnology BrownUniversity
Pasadena,CA91125 Providence,RI02912
USA USA
M.Golubitsky
DepartmentofMathematics
UniversityofHouston
Houston,TX77204-3476
USA
MathematicsSubjectClassification(2000):65-01,73V05,45L05,4601
LibraryofCongressCataloging-in-PublicationData
Atkinson,KendallE.
Theoreticalnumericalanalysis:afunctionalanalysisframework
/KendallAtkinson,WeiminHan.
p.cm.—(Textsinappliedmathematics;39)
Includesbibliographicalreferencesandindex.
ISBN0-387-95142-3(alk.paper)
1.Functionalanalysis. I.Han,Weimin. II.Title. IIISeries.
QA320.A852001
515—dc21 00–061920
Printedonacid-freepaper.
(cid:1)c 2001Springer-VerlagNewYork,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthe
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ISBN0-387-95142-3 SPIN10780644
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Dedicated to
Daisy and Clyde Atkinson
Hazel and Wray Fleming
and
Daqing Han, Suzhen Qin
Huidi Tang, Elizabeth Jing Han
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Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplines and a resurgence of interest in the modern as well as the clas-
sical techniques of applied mathematics. This renewal of interest, both in
research and teaching, has led to the establishment of the series: Texts in
Applied Mathematics (TAM).
Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevelof
excitement on the research frontier as newer techniques, such as numerical
and symbolic computer systems, dynamical systems, and chaos, mix with
and reinforce the traditional methods of applied mathematics. Thus, the
purpose of this textbook series is to meet the current and future needs of
these advances and encourage the teaching of new courses.
TAM will publish textbooks suitable for use in advanced undergraduate
and beginning graduate courses, and will complement the Applied Math-
ematical Sciences (AMS) series, which will focus on advanced textbooks
and research level monographs.
California Institute of Technology J.E. Marsden
Brown University L. Sirovich
University of Houston M. Golubitsky
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Preface
This textbook has grown out of a course that we teach periodically at the
University of Iowa. We have beginning graduate students in mathematics
whowishtoworkinnumericalanalysisfromatheoreticalperspective,and
they need a background in those “tools of the trade” that we cover in this
text. Ordinarily, such students would begin with a one-year course in real
and complex analysis, followed by a one- or two-semester course in func-
tional analysis and possibly a graduate level course in ordinary differential
equations, partial differential equations, or integral equations. We still ex-
pectourstudentstotakemostofthesestandardcourses,butwealsowant
to move them more rapidly into a research program. The course based on
this book is designed to facilitate this goal.
The textbook covers basic results of functional analysis and also some
additional topics that are needed in theoretical numerical analysis. Ap-
plications of this functional analysis are given by considering, at length,
numerical methods for solving partial differential equations and integral
equations.
Thematerialinthetextispresentedinamixedmanner.Sometopicsare
treated with complete rigor, whereas others are simply presented without
proof and perhaps illustrated (e.g., the principle of uniform boundedness).
We have chosen to avoid introducing a formalized framework for Lebesgue
measure and integration and also for distribution theory. Instead we use
standard results on the completion of normed spaces and the unique ex-
tension of densely defined bounded linear operators. This permits us to
introduce the Lebesgue spaces formally and without their concrete realiza-
tionusingmeasuretheory.Theweakderivativecanbeintroducedsimilarly