Five Mini-Courses on Analysis

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Transcription:

Christopher Heil Five Mini-Courses on Analysis Metrics, Norms, Inner Products, and Topology Lebesgue Measure and Integral Operator Theory and Functional Analysis Borel and Radon Measures Topological Vector Spaces October 25, 2010 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

Contents ANHA Series Preface.......................................... vii General Notation..............................................xxiii 1 Metrics, Norms, Inner Products, and Topology............ 1 1.1 Metrics and Convergence................................. 1 1.2 Norms and Seminorms................................... 2 1.2.1 Infinite Series in Normed Spaces..................... 4 1.2.2 Convexity........................................ 5 1.3 Examples of Banach Spaces............................... 6 1.3.1 The l p Spaces..................................... 6 1.3.2 Spaces of Continuous and Differentiable Functions..... 9 1.4 Inner Products.......................................... 12 1.5 Topology............................................... 16 1.5.1 Product Topologies................................ 18 1.6 Convergence and Continuity in Topological Spaces........... 19 1.6.1 Convergence...................................... 19 1.6.2 Continuity........................................ 22 1.6.3 Equivalent Norms................................. 24 1.7 Closed and Dense Sets................................... 26 1.8 Compactness............................................ 28 1.8.1 Compact Sets..................................... 28 1.8.2 Functions on Compact Sets......................... 30 1.8.3 Norms on Finite-Dimensional Spaces................. 31 1.9 Complete Sequences and a First Look at Schauder Bases..... 33 1.9.1 Span, Finite Independence, and Hamel Bases......... 34 1.9.2 Closed Span...................................... 35 1.9.3 Introduction to Schauder Bases..................... 37 1.9.4 Separable Banach Spaces........................... 38 1.10 Unconditional Convergence............................... 39 1.11 Orthogonality........................................... 42

vi Contents 1.11.1 Orthogonality and the Pythagorean Theorem......... 42 1.11.2 Orthogonal Direct Sums............................ 43 1.11.3 Orthogonal Projections and Orthogonal Complements. 43 1.12 Orthogonality and Complete Sequences.................... 45 1.13 Urysohn s Lemma....................................... 48 2 Lebesgue Measure and Integral............................ 51 2.1 Exterior Lebesgue Measure............................... 51 2.2 Lebesgue Measure....................................... 53 2.2.1 Definition and Basic Properties..................... 53 2.2.2 Equivalent Formulations of Measurability............. 56 2.2.3 Almost Everywhere................................ 58 2.2.4 Nonmeasurable Sets............................... 58 2.3 Measurable Functions.................................... 59 2.4 Convergence in Measure.................................. 62 2.5 The Lebesgue Integral................................... 63 2.5.1 Integration of Nonnegative Simple Functions.......... 63 2.5.2 Integration of Nonnegative Functions................ 64 2.5.3 The Monotone Convergence Theorem................ 65 2.5.4 Fatou s Lemma................................... 66 2.5.5 Integration of Real-Valued and Complex-Valued Functions........................................ 67 2.5.6 The Lebesgue Dominated Convergence Theorem...... 68 2.5.7 Relation to the Riemann Integral.................... 69 2.6 The L p Spaces.......................................... 70 2.6.1 Norm and Completeness........................... 70 2.6.2 On Abuses of Notation............................. 72 2.6.3 Convergence in L p (E).............................. 73 2.6.4 Dense Subsets of L p (E)............................ 74 2.6.5 The Hilbert Space L 2 (E)........................... 75 2.6.6 Local Integrability................................. 77 2.7 Repeated Integration.................................... 79 2.8 Functions of Bounded Variation........................... 81 2.8.1 Definition and Examples........................... 81 2.8.2 The Jordan Decomposition......................... 83 2.8.3 Differentiability of Functions of Bounded Variation.... 84 2.9 Singular Functions on the Real Line....................... 85 2.10 Absolutely Continuous Functions and the Banach Zarecki Theorem............................................... 87 2.10.1 Definition of Absolute Continuity.................... 87 2.10.2 Preparation for the Banach Zarecki Theorem......... 88 2.10.3 The Banach Zarecki Theorem....................... 91 2.11 The Lebesgue Differentiation Theorem..................... 93 2.12 The Fundamental Theorem of Calculus..................... 95

Contents vii 3 Operator Theory and Functional Analysis................. 99 3.1 Bounded Linear Operators on Normed Spaces............... 99 3.1.1 Equivalence of Bounded and Continuous Linear Operators........................................ 101 3.1.2 Isomorphisms..................................... 102 3.1.3 Eigenvalues and Eigenvectors....................... 103 3.1.4 Orthogonal Projections............................ 104 3.1.5 Multiplication Operators........................... 104 3.1.6 The Space B(X, Y )................................ 105 3.2 Integral Operators....................................... 108 3.2.1 Hilbert Schmidt Integral Operators.................. 110 3.2.2 Schur s Test...................................... 111 3.2.3 Convolution...................................... 112 3.3 Banach Algebras........................................ 112 3.4 The Dual of a Hilbert Space.............................. 114 3.5 The Dual of L p (E)...................................... 116 3.6 Adjoints of Operators on Hilbert Spaces.................... 117 3.6.1 Adjoints of Bounded Operators..................... 117 3.6.2 Adjoints of Unbounded Operators................... 118 3.6.3 Bounded Self-Adjoint Operators on Hilbert Spaces.... 119 3.6.4 Positive and Positive Definite Operators on Hilbert Spaces........................................... 121 3.7 Compact Operators on Hilbert Spaces...................... 123 3.7.1 Definition and Basic Properties..................... 123 3.7.2 Finite-Rank Operators............................. 125 3.7.3 Integral Operators with Square-Integrable Kernels..... 126 3.8 The Spectral Theorem for Compact Self-Adjoint Operators... 128 3.8.1 Existence of an Eigenvalue......................... 128 3.8.2 The Spectral Theorem............................. 129 3.9 Hilbert Schmidt Operators............................... 132 3.9.1 Definition and Basic Properties..................... 133 3.9.2 Singular Numbers and Schatten Classes.............. 134 3.9.3 Trace-Class Operators............................. 136 3.9.4 Hilbert Schmidt Integral Operators.................. 137 3.10 The Hahn Banach Theorem.............................. 139 3.10.1 Abstract Statement of the Hahn Banach Theorem..... 140 3.10.2 Corollaries of the Hahn Banach Theorem............. 140 3.10.3 Orthogonal Complements in Normed Spaces.......... 142 3.10.4 X and Reflexivity............................... 143 3.10.5 Adjoints of Operators on Banach Spaces............. 144 3.11 The Baire Category Theorem............................. 145 3.12 The Uniform Boundedness Principle....................... 147 3.13 The Open Mapping Theorem............................. 150 3.14 The Closed Graph Theorem.............................. 152 3.15 Schauder Bases......................................... 153

viii Contents 3.15.1 Continuity of the Coefficient Functionals............. 153 3.15.2 Minimal Sequences................................ 155 3.15.3 A Characterization of Schauder Bases................ 156 3.15.4 Unconditional Bases............................... 157 3.16 Weak and Weak* Convergence............................ 158 4 Borel and Radon Measures on the Real Line.............. 161 4.1 σ-algebras............................................. 161 4.2 Signed Measures........................................ 163 4.2.1 The Jordan Decomposition......................... 165 4.3 Positive Measures and Integration......................... 168 4.3.1 Basic Properties of Positive Measures................ 168 4.3.2 Borel Measurable Functions........................ 168 4.3.3 Integration of Nonnegative Functions................ 169 4.3.4 Integration of Arbitrary Functions................... 170 4.4 Signed Measures and Integration.......................... 172 4.5 Complex Measures....................................... 175 4.6 Fubini and Tonelli for Borel Measures...................... 179 4.7 Radon Measures......................................... 180 4.8 The Riesz Representation Theorem for Positive Functionals on C c (R)............................................... 182 4.8.1 Topologies on C c (R)............................... 183 4.8.2 Positive Linear Functionals on C c (R)................ 184 4.9 The Relation Between Radon and Borel Measures........... 186 4.10 The Dual of C 0 (R)...................................... 188 5 Topological Vector Spaces................................. 191 5.1 Motivation and Examples................................ 191 5.2 Topological Vector Spaces................................ 196 5.2.1 Base for a Topology............................... 196 5.2.2 Topological Vector Spaces.......................... 197 5.3 Topologies Induced by Families of Seminorms............... 198 5.3.1 Motivation....................................... 198 5.3.2 The Topology Associated with a Family of Seminorms. 199 5.3.3 The Convergence Criterion......................... 200 5.3.4 Continuity of the Vector Space Operations............ 202 5.3.5 Continuity Equals Boundedness..................... 203 5.4 Topologies Induced by Countable Families of Seminorms..... 205 5.4.1 Metrizing the Topology............................ 205 5.5 Tempered and Compactly Supported Distributions.......... 207 5.5.1 Tempered Distributions............................ 207 5.5.2 Compactly Supported Distributions.................. 208 5.6 Cc (R) and its Dual Space D (R).......................... 209 5.6.1 The Topology on Cc (R)........................... 210 5.6.2 The Space of Distributions......................... 212

Contents ix 5.6.3 Inclusions........................................ 214 5.7 The Weak and Weak* Topologies on a Normed Linear Space.. 217 5.7.1 The Weak Topology............................... 218 5.7.2 The Weak* Topology.............................. 219 5.8 Alaoglu s Theorem...................................... 220 5.8.1 Product Topologies................................ 220 5.8.2 Tychonoff s Theorem.............................. 221 5.8.3 Statement and Proof of Alaoglu s Theorem........... 221 5.8.4 Implications for Separable Spaces.................... 223 A Zorn s Lemma............................................. 227 Hints for Exercises and Additional Problems.................. 229 Index of Symbols.............................................. 245 References..................................................... 251 Index.......................................................... 255

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