Linear Functional Analysis: An Application-Oriented Introduction

446 Pages·2016·3.448 MB·English

by  Hans Wilhelm Alt, Robert Nürnberg

Universitext
Hans Wilhelm Alt
Linear
Functional
Analysis
An Application-Oriented Introduction
Translated by Robert Nürnberg
Universitext
Universitext
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Hans Wilhelm Alt
Linear Functional Analysis
An Application-Oriented Introduction
Translated by Robert Nürnberg
123
Hans Wilhelm Alt
Technische Universität München
Garching near Munich
Germany
Translation from German language edition:
Lineare Funktionalanalysis
by Hans Wilhelm Alt
Copyright © 2012, Springer Berlin Heidelberg
Springer Berlin Heidelberg is part of Springer Science + Business Media
All Rights Reserved
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-1-4471-7279-6 ISBN978-1-4471-7280-2 (eBook)
DOI10.1007/978-1-4471-7280-2
LibraryofCongressControlNumber:2016944464
Mathematics Subject Classification: 46N20, 46N40, 46F05, 47B06, 46G10
©Springer-Verlag London 1985, 1991, 1999, 2002, 2006, 2012, 2016
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Preface
ThepresentbookistheEnglishtranslationofapreviousGermanedition,
alsopublishedbySpringerVerlag.ThetranslationwascarriedoutbyRobert
Nu¨rnberg, who also did a marvellous job at detecting errors and mistakes in
the original version. In addition, Andrei Iacob revised the English version.
The book originated in a series of lectures I gave for the first time at the
University of Bochum in 1980, and since then it has been repeatedly used in
many lectures by me and other mathematicians and during this time it has
changedaccordingly. IprovidethereaderwithanintroductiontoFunctional
Analysis as a synthesis of Algebra, Topology, and Analysis, which is the
source for basic definitions which are important for differential equations.
The book includes a number of appendices in which special subjects are
presented in more detail. Therefore its content is rich enough for a lecturer
to find enough material to fill a course in functional analysis according to
his special interests. The text can also be used as an additional source for
lectures on partial differential equations or advanced numerical analysis.
It must be said that my strategy has been dictated by the desire to offer
the reader an easy and fast access to the main theorems of linear functional
analysis and, at the same time, to provide complete proofs. So there is a
separate appendix where the Lebesgue integral is introduced in a complete
functional analytic way, andanappendix whith details for Sobolev functions
which complete the proofs of the embedding theorems. Therefore the text is
self-contained and the reader will benefit from this fact.
Parallel to this edition, a revised German version has become available
(Lineare Funktionalanalysis, 6. Edition, Springer 2012) with the same math-
ematical content. This is made possible by a common source text. Therefore
one does not have to worry about the content in different versions. I am
happy that this book is now accessible to a wider community.
If you find any errors or misprints in the text, please point them out to
theauthorviaemail:“[email protected]”. Thiswillhelptoimprovethetextof
possible future editions.
I hope that this book is written in the good tradition of functional anal-
ysis and will serve its readers well. I thank Springer Verlag for making the
publication of this edition possible and for their kind support over many
years.
Technical University Munich, August 2015
H. W. Alt
V

Table of Contents
1 Introduction.............................................. 1
2 Preliminaries ............................................. 9
2.1 Scalar product ..................................... 9
2.3 Orthogonality ...................................... 11
2.4 Norm ............................................. 13
2.6 Metric ............................................ 16
2.8 Examples of metrics ................................ 16
2.9 Balls and distance between sets ...................... 18
2.10 Open and closed sets............................... 19
2.11 Topology ......................................... 19
2.14 Comparison of topologies ........................... 21
2.15 Comparison of norms .............................. 21
2.17 Convergence and continuity......................... 23
2.18 Convergence in metric spaces ....................... 24
2.21 Completeness ..................................... 27
2.22 Banach spaces and Hilbert spaces.................... 27
2.23 Sequence spaces ................................... 28
2.24 Completion ....................................... 30
E2 Exercises .............................................. 31
E2.6 Completeness of Euclidean space .................... 34
E2.7 Incomplete function space .......................... 34
E2.9 Hausdorff distance between sets..................... 35
3 Function spaces........................................... 37
3.1 Bounded functions.................................. 37
3.2 Continuous functions on compact sets ................. 38
3.3 Continuous functions................................ 39
3.4 Support of a function ............................... 41
3.5 Differentiable functions.............................. 41
3.7 H¨older continuous functions.......................... 44
3.9 Measures .......................................... 45
3.10 Examples of measures.............................. 46
3.11 Measurable functions............................... 47
VII
VIII Table of Contents
3.15 Lebesgue spaces ................................... 50
3.18 H¨older’s inequality................................. 52
3.19 Majorant criterion in Lp............................ 55
3.20 Minkowski inequality............................... 55
3.21 Fischer-Riesz theorem.............................. 55
3.23 Vitali’s convergence theorem ........................ 57
3.25 Lebesgue’s general convergence theorem .............. 60
3.27 Sobolev spaces .................................... 63
E3 Exercises .............................................. 66
E3.3 Standard test function ............................. 67
E3.4 Lp-norm as p→∞ ................................ 67
E3.6 Fundamental theorem of calculus.................... 68
A3 Lebesgue’s integral ..................................... 71
A3.3 Elementary Lebesgue measure ...................... 72
A3.4 Outer measure.................................... 73
A3.5 Step functions .................................... 74
A3.6 Elementary integral ............................... 75
A3.8 Lebesgue integrable functions....................... 78
A3.10 Axioms of the Lebesgue integral ................... 79
A3.14 Integrable sets ................................... 84
A3.15 Measure extension ............................... 87
A3.18 Egorov’s theorem ................................ 90
A3.19 Majorant criterion ............................... 91
A3.20 Fatou’s lemma................................... 93
A3.21 Dominated convergence theorem ................... 94
4 Subsets of function spaces ................................ 95
4.1 Convexity ......................................... 95
4.3 Projection theorem ................................. 96
4.5 Almost orthogonal element .......................... 99
4.6 Compactness....................................... 100
4.12 Arzel`a-Ascoli theorem (compactness in C0) ........... 106
4.13 Convolution ...................................... 107
4.14 Dirac sequences ................................... 110
4.16 Riesz theorem (compactness in Lp) .................. 112
4.18 Examples of separable spaces ....................... 115
4.19 Cut-off function ................................... 118
4.20 Partition of unity.................................. 118
4.22 Fundamental lemma of calculus of variations .......... 122
4.23 Local approximation of Sobolev functions............. 122
4.25 Product rule for Sobolev functions ................... 124
4.26 Chain rule for Sobolev functions..................... 125
E4 Exercises .............................................. 126
E4.4 Strictly convex spaces.............................. 128
E4.5 Separation theorem in IRn.......................... 129
Table of Contents IX
E4.6 Convex functions.................................. 129
E4.7 Characterization of convex functions................. 131
E4.8 Supporting planes ................................. 132
E4.9 Jensen’s inequality ................................ 133
E4.11 The space Lp for p<1............................ 134
E4.13 Compact sets in (cid:2)2 ............................... 135
E4.15 Comparison of Ho¨lder spaces ...................... 136
E4.16 Compactness with respect to the Hausdorff metric.... 137
E4.18 Continuous extension ............................. 138
E4.19 Dini’s theorem................................... 139
E4.20 Nonapproximability in C0,α ....................... 139
E4.21 Compact sets in Lp............................... 139
5 Linear operators.......................................... 141
5.2 Linear operators.................................... 142
5.7 Neumann series .................................... 146
5.8 Theorem on invertible operators...................... 147
5.9 Analytic functions of operators ....................... 147
5.10 Examples (exponential function)..................... 148
5.12 Hilbert-Schmidt integral operators ................... 149
5.14 Linear differential operators......................... 151
5.17 Distributions (The space D(cid:2)(Ω)) .................... 152
5.20 Topology on C∞(Ω) ............................... 156
0
5.21 The space D(Ω)................................... 157
E5 Exercises .............................................. 160
E5.3 Unique extension of linear maps..................... 160
E5.4 Limit of linear maps............................... 161
6 Linear functionals ........................................ 163
6.1 Riesz representation theorem......................... 163
6.2 Lax-Milgram theorem ............................... 164
6.4 Elliptic boundary value problems ..................... 167
6.5 Weak boundary value problems....................... 169
6.6 Existence theorem for the Neumann problem........... 170
6.7 Poincar´e inequality ................................. 171
6.8 Existence theorem for the Dirichlet problem ........... 171
6.10 Variational measure................................ 173
6.11 Radon-Nikody´m theorem ........................... 173
6.12 Dual space of Lp for p<∞ ......................... 175
6.14 Hahn-Banach theorem ............................. 180
6.15 Hahn-Banach theorem (for linear functionals) ......... 182
6.20 Spaces of additive measures......................... 185
6.21 Spaces of regular measures.......................... 185
6.23 Riesz-Radon theorem .............................. 187
6.25 Functions of bounded variation...................... 191