Table Of ContentCompact Textbooks in Mathematics
Christian Clason
Introduction
to Functional
Analysis
Compact Textbooks in Mathematics
CompactTextbooksinMathematics
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Christian Clason
Introduction to
Functional Analysis
ChristianClason
Fakulta¨tfu¨rMathematik
Universita¨tDuisburg-Essen
Essen,Germany
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-030-52783-9 ISBN978-3-030-52784-6 (eBook)
https://doi.org/10.1007/978-3-030-52784-6
MathematicsSubjectClassification:46-01
©SpringerNatureSwitzerlandAG2020
Translation from the German language edition: “Einführung in die Funktionalanalysis” by Christian
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Preface
Functionalanalysisisthecontinuationoflinearalgebraby
othermeans.
Thedevelopmentoffunctionalanalysisintheearlytwentiethcenturywasmotivated
bythedesireforgeneralresultsonthesolvabilityofdifferentialequations.Instead
ofsolvingconcretedifferentialequationslikef(cid:2)(cid:2)+f =gforagivengbyspecific
techniques, people wanted to know which properties of a differential equation or
aright-handsideg werenecessaryforsuchasolutiontoexist.Thecrucialinsight
here was to consider functionsas pointsin a vector space, on which the mapping
D : f (cid:3)→ f(cid:2)(cid:2) + f defines a linear differential operator. A similar step from
concrete systems of linear equations to the abstract linear equation Ax = b for
a matrix A and a vector b is the basis of linear algebra. The question was then
about the properties of D required for unique solvability of Df = g, in analogy
to the injectivity and surjectivity of A or the absence of 0 as an eigenvalue of A.
Here, the main difficulty lies in the fact that many of the fundamental results of
linear algebraare based on the finite-dimensionalityof the involvedvectorspaces
(e.g.,usingtherank-nullitytheorem).However,thisisnolongerthecaseforvector
spacesoffunctions,anditbecomesnecessarytoconsiderthesealgebraicconcepts
in combination with topological concepts such as convergence and compactness.
One of the central themes of this book is to work out which algebraic, metric,
topological,andgeometricpropertiescanserveassubstitutesforthemissingfinite-
dimensionality,andwhatrolepreciselytheseplayfortheindividualresults.Thefact
that this combination leads to extremely rich structures is what makes functional
analysissoappealingandhasledtoitbecominganessentialfoundationformodern
appliedmathematics,fromthetheoryandnumericalsolutionofpartialdifferential
equations, through optimization and probability theory, to medical imaging and
mathematicalimageprocessing.
The contents of this book correspondexactly to those of 26 lectures of 90min
each(ofwhichnottoomanyshouldfallonapublicholiday)inthefourthsemester
of a bachelor’s program in mathematics; it therefore cannot and is not supposed
v
vi Preface
to be a substitute for more comprehensive textbooks such as [7,15,22,29–31]
or the more recent [3,23] (which are closer to the aims of this book). Its aim is
ratherto presenta concise,streamlined,andrigorousdevelopmentofthe essential
structural results that are important in particular throughout applied mathematics
andthusbuildasolidfoundationformathematicallecturesontheabove-mentioned
topics.Allfurtherresults(e.g.,onquotientspacesorFredholmoperators)aretreated
onlyinsofar as they are neededor are usefulfor significantlysimplifyinga proof.
After the tight corset of a lecture had been removed,many more beautiful results
couldhavebeenincluded;thistemptationwasresisted.Readersarethereforecertain
to miss a favorite result; a particularly conspicuous gap is details on Lebesgue
and Sobolev spaces, which are left to lectures on measure and integration theory
(e.g., based on [5, Chapters XII and XIII]) and on partial differential equations,
respectively.(Anicetreatmentcanalsobefoundin[3].)
Thestructureofthebookaswellisbasedonthedesiretodrawasclearanddirect
a lineaspossibleto themainresults.To thisend,asfar aspossible,relatedtopics
are treated together, and more general results are exploited that are to be proved
anyway.Thisleadstonumerousdependenciesbutislikelyinthespiritoffunctional
analysisasanabstractstructuraltheory(motivatedbyapplicationinotherbranches
of mathematics). Of course, some freedom remains even under these constraints
that others might have used differently, in particular for an earlier treatment of
Hilbert spaces. Specifically, it is possible to cover Chap.15 directly after Chap.3,
with the exception of the Lax–Milgram theorem (Theorem 15.11, which requires
Corollary 9.8) and the Fischer–Riesz theorem (Corollary 15.16, which requires
the conceptof isomorphismfromChap.4).Chapter 16requiresthe Hahn–Banach
theorem (Theorem 8.1); the concept of a Hilbert-space adjoint operator is here
definedviathatofthe(Banachspace)adjointoperatorbutcouldalsobeintroduced
independently.Finally,Chap.17continuesseamlesslyfromChap.14.
The chosen arrangement also corresponds to the mathematical tradition of
presentinga theorywithoutanyreferencetoitshistoricaldevelopment.Thisoften
allowsaclearerviewofthecentralstructuresandaconcentrationonconceptsthat
have proven to be particularly fruitful. Conversely, one risks losing sight of the
fact that these structures and concepts were developed by people who now only
appear as eponyms for theorems. In particular, those are neglected who in initial
phasesproducedgroundbreakingresultsthatwerelatersubsumedinmoregeneral
statements. This is especially true for functional analysis, which was brought to
essentially the state as presentedhere by a comparativelysmall numberof people
withinafewdecades(betweenabout1900and1940).Ratherthanoverloadingthis
bookwith(evenmore)footnotes,thereaderisreferredtotheliteratureonthispoint.
Anextensiveaccountofthehistoryoffunctionalanalysiscanbefoundin[9],ashort
summaryof whichis contained(andmore easily available)as an appendixin [2].
Thedevelopmentoftheresultstreatedinthisbookisdescribedmoreaccessibly—
andmoreconversationally—in[12,PartXIX].Anevenmoreextensiveaccountup
Preface vii
to the present is given in [18]; this work also contains a detailed chronology as
well as numerousquotations from and references to original works that are often
still worth reading. Historical and biographical remarks can also be found in the
appendicestotheindividualchaptersof[28](whicharealsoworthreading).
Thisbookisbasedmainlyon[1,4,14,27,28],whicharealsorecommendedfor
supplementalreading.ThanksareduetoMartinBrokateandGerdWachsmuthfor
their lecture notes and for helpful comments, Otmar Scherzer and Anton Schiela
forhelpfulcommentsaswell,andRemoKretschmannforhelpwiththeexercises.
Finally,DavidP.Kramer’sexpertcopy-editingwasmuchappreciated.
Essen,Germany ChristianClason
May2020
Contents
PartI TopologicalBasics
1 MetricSpaces ............................................................... 3
2 CompactSets................................................................ 9
PartII LinearOperatorsBetweenNormedSpaces
3 NormedVectorSpaces..................................................... 19
4 LinearOperators........................................................... 35
5 TheUniformBoundednessPrinciple..................................... 45
6 QuotientSpaces............................................................. 55
PartIII DualSpacesandWeakConvergence
7 LinearFunctionalsandDualSpaces ..................................... 63
8 TheHahn–BanachTheorem .............................................. 71
9 AdjointOperators.......................................................... 83
10 Reflexivity ................................................................... 93
11 WeakConvergence ......................................................... 99
PartIV CompactOperatorsBetweenBanachSpaces
12 CompactOperators ........................................................ 111
13 TheFredholmAlternative................................................. 119
14 TheSpectrum............................................................... 123
ix
x Contents
PartV HilbertSpaces
15 InnerProductsandOrthogonality ....................................... 135
16 TheRieszRepresentationTheorem ...................................... 149
17 SpectralDecompositioninHilbertSpaces............................... 157
References......................................................................... 165
Index............................................................................... 167
