Contents
Preface ...................................... xi
1 Classical Abstract Spaces in Functional Analysis
1.1 IntroductionandNotation....................... 1
1.2 TopologicalSpaces........................... 5
1.2.1 ConvergenceinTopologicalSpaces.............. 13
1.2.2 Continuity of Functions on Topological Spaces . . . . . . . 15
1.2.3 Weak Topology . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.4 Compactness of Sets in Topological Spaces . . . . . . . . . 19
1.3 MetricSpaces.............................. 21
1.3.1 Convergence and Continuity in Metric Spaces . . . . . . . . 21
1.3.2 Closed and Dense Sets in Metric Spaces . . . . . . . . . . . 23
1.3.3 CompleteMetricSpaces.................... 23
1.3.4 TheBaireCategoryTheorem................. 25
1.3.5 CompactnessofSetsinMetricSpaces ............ 27
1.3.6 Equicontinuous Functions on Metric Spaces . . . . . . . . . 30
1.3.7 TheArzela-AscoliTheorem.................. 33
1.3.8 H¨older’s and Minkowski’s Inequalities . . . . . . . . . . . . 35
1.4 VectorSpaces.............................. 41
1.5 NormedVectorSpaces......................... 45
1.5.1 BasicDefinitions........................ 45
1.5.2 ExamplesofNormedVectorSpaces ............. 46
1.6 Space of Lebesgue Measurable Functions . . . . . . . . . . . . . . . 52
1.6.1 IntroductiontoMeasureTheory ............... 52
1.6.2 LebesgueIntegral........................ 54
1.6.3 MeasurableFunctions ..................... 57
1.7 HilbertSpaces ............................. 58
2 Linear Functionals and Linear Operators
2.1 Fundamental Theorems of Analysis . . . . . . . . . . . . . . . . . . 65
2.1.1 Hahn-BanachTheorem .................... 65
2.1.2 Uniform Boundedness Theorem . . . . . . . . . . . . . . . . 69
2.1.3 TheOpenMappingTheorem................. 71
2.2 DualSpaces............................... 75
2.3 TheWeakTopology .......................... 79
2.4 The Weak* Topology.......................... 80
2.5 SignedMeasuresandTopology .................... 88
2.6 Riesz’sRepresentationTheorem.................... 91
2.6.1 Space of Lebesgue Measurable Functions . . . . . . . . . . . 91
2.6.2 HilbertSpaces ......................... 94
2.7 ClosedOperatorsonHilbertSpaces ................. 95
2.8 AdjointOperators ........................... 97
2.9 GelfandTriples............................. 103
2.10 Bilinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Common Function Spaces in Applications
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3.1 The L Spaces ............................. 111
3.2 SobolevSpaces ............................. 113
3.2.1 DistributionalDerivatives................... 114
3.2.2 Sobolev Spaces, Integer Order . . . . . . . . . . . . . . . . . 117
3.2.3 Sobolev Spaces, Fractional Order . . . . . . . . . . . . . . . 118
3.2.4 TraceTheorems ........................ 122
3.2.5 The Poincar′eInequality.................... 123
3.3 Banach Space Valued Functions . . . . . . . . . . . . . . . . . . . . 126
3.3.1 BochnerIntegrals........................ 126
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3.3.2 The Space L (0,T),X .................... 131
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3.3.3 The Space W (0,T),X .................. 133
4 Di?erential Calculus in Normed Vector Spaces
4.1 Di?erentiability of Functionals . . . . . . . . . . . . . . . . . . . . 137
4.1.1 Gateaux Di?erentiability . . . . . . . . . . . . . . . . . . . 137
4.1.2 Fr′echet Di?erentiability . . . . . . . . . . . . . . . . . . . . 139
4.2 Classical Examples of Di?erentiable Operators . . . . . . . . . . . 143
5 Minimization of Functionals
5.1 TheWeierstrassTheorem ....................... 161
5.2 ElementaryCalculus.......................... 163
5.3 Minimization of Di?erentiable Functionals . . . . . . . . . . . . . . 165
5.4 EqualityConstrainedSmoothFunctionals.............. 166
5.5 Fr′echetDi?erentiableImplicitFunctionals.............. 171
6 Convex Functionals
6.1 CharacterizationofConvexity..................... 177
6.2 Gateaux Di?erentiable Convex Functionals . . . . . . . . . . . . . 180
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6.3 Convex Programming in R ...................... 183
6.4 OrderedVectorSpaces......................... 188
6.4.1 Positive Cones, Negative Cones, and Orderings . . . . . . . 189
6.4.2 OrderingsonSobolevSpaces ................. 191
6.5 Convex Programming in Ordered Vector Spaces . . . . . . . . . . . 193
6.6 Gateaux Di?erentiable Functionals on Ordered Vector Spaces . . . 199
Lower Semicontinuous Functionals
7.1 CharacterizationofLowerSemicontinuity .............. 205
7.2 Lower Semicontinuous Functionals and Convexity . . . . . . . . . . 208
7.2.1 Banach Theorem for Lower Semicontinuous Functionals . . 208
7.2.2 Gateaux Di?erentiability . . . . . . . . . . . . . . . . . . . 210
7.2.3 Lower Semicontinuity in Weak Topologies . . . . . . . . . . 210
7.3 TheGeneralizedWeierstrassTheorem ................ 212
7.3.1 CompactnessinWeakTopologies............... 213
7.3.2 Bounded Constraint Sets . . . . . . . . . . . . . . . . . . . 215
7.3.3 Unbounded Constraint Sets . . . . . . . . . . . . . . . . . . 215
7.3.4 ConstraintSetsonOrderedVectorSpaces.......... 217
eferences .................................... 221
ndex ....................................... 223