Functional analysis: an introductory course

210 Pages·2018·1.5 MB·English

by  Ovchinnikov

Universitext
Sergei Ovchinnikov
Functional
Analysis
An Introductory Course
Universitext
Universitext
Series editors
Sheldon Axler
San Francisco State University
Carles Casacuberta
Universitat de Barcelona
Angus MacIntyre
Queen Mary University of London
Kenneth Ribet
University of California, Berkeley
Claude Sabbah
École polytechnique, CNRS, Université Paris-Saclay, Palaiseau
Endre Süli
University of Oxford
Wojbor A. Woyczyński
Case Western Reserve University
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Sergei Ovchinnikov
Functional Analysis
An Introductory Course
123
SergeiOvchinnikov
Department ofMathematics
SanFrancisco State University
SanFrancisco, CA
USA
ISSN 0172-5939 ISSN 2191-6675 (electronic)
Universitext
ISBN978-3-319-91511-1 ISBN978-3-319-91512-8 (eBook)
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Preface
Therearemanyexcellenttextbooksonfunctionalanalysis.However,onlyfewcan
be regarded as introductory or elementary. In writing this book, my goal was to
produce a “really” introductory (and short) text on this subject.
This book originated from the notes that I kept while teaching the graduate
courseIntroductoryFunctionalAnalysisatSanFranciscoStateUniversity(SFSU).
The Mathematics Department of SFSU offers the Master of Arts in Mathematics
degree to its students. Many students who received an MA degree from this
departmentdonotpursueahigherdegree. WhenteachingthiscourseIwantedmy
studentstogetabetterfeelforanalysisingeneral,appreciateitsimportance,andbe
readytolearnmoreaboutit,shouldtheneedarise.Someofthesestudentshadgaps
in their knowledge of basic analysis and other branches of mathematics. For this
reason, in addition to functional analysis topics, the book contains materials that
any undergraduate/graduate student of mathematics should know. These materials
are subjects of the first and second chapters.
The first chapter presents selected topics of general mathematical interest that I
feltareneededtomasterthematerialpresentedinthebook.First,itincludesabrief
overviewofconvexfunctionswithapplicationstoclassicalinequalities,thatwillbe
used inthe succeedingchapters. Second, binaryrelations of equivalence and order
and their main properties are described. The chapter ends with Zorn’s lemma,
which is essential in the establishing some advanced results later in the book.
Most spaces in functional analysis are topological (often metric) vector spaces.
Thisiswhynaturalprerequisitesforacourseinfunctionalanalysisincludecourses
inlinearalgebraandrealanalysis.Idonotincludeapreliminaryoverviewoflinear
algebra. However, I felt that it is important to review basics of metric and topo-
logicalspaces tofill thegapsbetween what the reader probablylearned some time
ago and whatis necessarytomastertopicscovered inthe book. Thisreviewisthe
subject of the second chapter. In addition to standard topics on metric spaces
coveredinrealanalysis,Iprovea“light”versionoftheBaireCategoryTheoremin
Chapter 2. A brief discussion of this theorem is found at the end of the chapter.
vii
viii Preface
Metric and topological vector spaces are introduced in the first section of
Chapter 3. The rest of this chapter is devoted to “special spaces” that serve as
illustrations to many general theorems in the book. These spaces are
finite-dimensional spaces ‘np, sequence spaces ‘p, c, c0, and s, and spaces of con-
tinuousfunctionsandfunctionsofboundedvariation.BecauseIwantedtokeepthis
book on an “introductory” level, I did not include in this chapter spaces of
Lebesgue measurable functions despite their great role in functional analysis. The
main thrust of this chapter is on establishing properties such as completeness and
separability of the spaces under consideration.
Normed spaces, bounded linear operators on these spaces, and their properties
are subjects of Chapter 4. The concept of a Schauder basis (cf. Section 4.1) illus-
trates the advantage of blending the metric and linear structures of normed spaces
allowing for formation of convergent series. The central result of Section 4.2 is
equivalence of continuity and boundedness properties of linear operators, thus
taking an advantage of the vector space structure in establishing analytical prop-
erties of continuous linear operators. Fundamental topological properties of
finite-dimensional normed spaces are discussed in Sections 4.3 and 4.4.
A separate chapter in the book is devoted to linear functionals. First, in
Section5.1,itisdemonstratedthatmanynormedspaceshaveconsiderably“large”
dualspaces,thatis,spacesofcontinuouslinearfunctionalsontheunderlyingspace.
In Sections 5.2 and 5.3 various results known as the Hahn–Banach Theorem are
established. This theorem plays a fundamental role in functional analysis. In par-
ticular,theHahn–BanachTheoremshowsthatthereare“enough”continuouslinear
functionals on a normed space to motivate the study of the dual space. As an
application of this theorem, all bounded linear functionals on the space C½a;b(cid:2) are
describedinSection5.4.InconclusionofChapter5,reflexivespacesareintroduced
and their elementary properties established.
Three “pillars” of functional analysis—the Uniform Boundedness, Open
Mapping, and Closed Graph Theorems—are covered in Chapter 6. (The forth
“pillar”istheHahn–BanachTheorem(cf.Chapter5)).Thedistinguishedroleofthe
completeness property in establishing these fundamental results is illustrated by a
number of examples and counterexamples in Section 6.4.
It is hard to imagine a textbook on functional analysis that does not include
topicsonHilbertspaces.Chapter7(thatconstitutesabout20%oftheentirebook)is
aninintroductiontothetheoryofHilbertspaces.Itbeginswithashortcoverageof
innerproductspacesinSection7.1.Thenexttwosectionsintroduceanddiscussthe
orthogonality property in inner product spaces and orthonormal families in Hilbert
spaces. In my opinion, the concept of summable families of vectors in normed
spacesiselementaryenoughtobeincludedintoanintroductorycourseinfunctional
analysis. This notion is used in Section 7.3 to define Hilbert bases and establish
convenient representation of vectors in Hilbert spaces. In the same section, the
Gram–Schmidtprocess isused toconstruct orthonormalbases inseparableHilbert
spaces. Effective tools for studying linear functionals and operators on Hilbert
spacesaretheRieszRepresentationtheoremandsesquilinearformsthatarecovered
Preface ix
in Sections 7.4 and 7.5, respectively. The concept of Hilbert-adjoint operator (cf.
Section 7.6) is used to introduce the standard classes of normal, self-adjoint, and
unitaryoperatorsinSection7.7.Projectionoperatorsarecoveredinthelastsection
of this chapter.
To enhance the presentation of Hilbert spaces, more advanced topics on sum-
mablefamiliesofvectorsinanormedspaceareincludedintheAppendixA,where
Hilbert spaces ‘ ðJÞ are described for arbitrary set J.
2
As I mentioned at the beginning of this Preface, I wanted to write an intro-
ductory text on functional analysis. In fact, I wanted to keep the book on “ele-
mentary”level.Forthisreason,thebookdoesnotrelyontheLebesguetheoryand
does not include the Spectral theorem. However, some topics that are usually
considered as not completely elementary are included in the book. For instance,
Zorn’slemma,Baire’sCategorytheoremandsummabilityoffamiliesofvectorsin
a normed space are included in the book.
Inmyopinion,themost effectivewayoflearningmathematics isby“doing it”.
There are 240 exercises in the book. Most exercises are “proof” problems, that is,
thereaderisinvitedtoproveastatementintheexercise.Everychapterhasitsown
set of exercises.
Throughout the book the symbol F denotes either the real field, R, or the
complex field, C. Symbols N, Z, and Q denote the sets of natural numbers
f1;2;...g, integers, and rational numbers, respectively. The symbol (cid:3) is always
used for the set inclusion relation, proper or not.
I wish to thank Sheldon Axler for his endorsement of this project and an
anonymous referee for reading the manuscript carefully and suggesting mathe-
matical and stylistic corrections. My special thanks go to Eric Hayashi for
numerouscommentsandrecommendationswhichmateriallyimprovedtheoriginal
draft of the book.
Berkeley, California Sergei Ovchinnikov
March 2018
Contents
1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Convex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Equivalence Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Zorn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Metrics and Pseudometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Open and Closed Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Convergence and Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Completion of a Metric Space. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 The Baire Category Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Topological Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Special Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Metric Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Finite-Dimensional ‘p Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Sequence Spaces ‘p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Sequence Spaces c, c , and c . . . . . . . . . . . . . . . . . . . . . . . . . . 56
0 00
3.5 The Sequence Space s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Spaces of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7 The BV½a;b(cid:2) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Properties of Normed Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Linear Operators and Functionals . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Finite-Dimensional Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Compactness in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 92
xi