Bruce K. Driver Contents Analysis Tools with Applications October 30, 2003 File:anal.tex Part I Back Ground Material 1 Introduction / User Guide ================================= 3 1.1 Topologybeginnings..................................... 3 1.2 A Better Integral and an Introduction to Measure Theory .... 3 2 Set Operations ============================================ 7 2.1 Exercises............................................... 10 3 The Real and Complex Numbers ========================== 11 3.1 TheRealNumbers ...................................... 12 3.1.1 TheDecimalRepresentationofaRealNumber........ 16 3.2 TheComplexNumbers................................... 19 3.3 Exercises............................................... 20 4 Limits and Sums =========================================== 21 4.1 Limsups,LiminfsandExtendedLimits..................... 21 4.2 Sumsofpositivefunctions................................ 24 4.3 Sumsofcomplexfunctions................................ 28 4.4 IteratedsumsandtheFubiniandTonelliTheorems.......... 32 4.5 Exercises............................................... 34 4.5.1 LimitProblems................................... 34 4.5.2 DominatedConvergenceTheoremProblems .......... 35 5 `p — spaces, Minkowski and Holder Inequalities ============ 39 5.1 Exercises .............................................. 44 Springer Part II Metric and Banach Space Basics Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo 4Contents Contents 5 6 Metric Spaces ============================================= 49 11 Compactness ==============================================135 6.1 Continuity.............................................. 51 11.1MetricSpaceCompactnessCriteria........................136 6.2 CompletenessinMetricSpaces............................ 53 11.2CompactOperators......................................142 6.3 Supplementary Remarks .................................. 55 11.3 Local and —Compactness...............................144 6.3.1 WordofCaution.................................. 55 11.4FunctionSpaceCompactnessCriteria......................146 6.3.2 RiemannianMetrics............................... 56 11.5 Tychono’sTheorem ....................................149 6.4 Exercises............................................... 57 11.6Exercises...............................................152 11.6.1Ascoli-ArzelaTheoremProblems....................153 7 Banach Spaces ============================================= 61 11.6.2 Tychono’sTheoremProblem......................154 7.1 Examples............................................... 61 7.2 BoundedLinearOperatorsBasics ......................... 64 7.3 GeneralSumsinBanachSpaces........................... 68 Part IV Topological Spaces II 7.4 Inverting Elements in O([) ............................... 70 7.5 HahnBanachTheorem................................... 71 12 Locally Compact Hausdor Spaces ========================157 7.6 Exercises............................................... 75 12.1LocallycompactformofUrysohn’sMetrizationTheorem.....161 7.6.1 Hahn-BanachTheoremProblems.................... 77 12.2PartitionsofUnity ......................................164 12.3 F0([) and the Alexanderov Compactification...............169 8TheRiemannIntegral===================================== 79 12.4 More on Separation Axioms: Normal Spaces ................171 8.1 TheFundamentalTheoremofCalculus..................... 83 12.5Stone-WeierstrassTheorem...............................174 8.2 Integral Operators as Examples of Bounded Operators ....... 85 12.6LocallyCompactVersionofStone-WeierstrassTheorem......178 8.3 Linear Ordinary DierentialEquations..................... 87 12.7Exercises...............................................179 8.4 ClassicalWeierstrassApproximationTheorem............... 91 8.5 Iterated Integrals ........................................ 96 13 Baire Category Theorem ==================================183 8.6 Exercises............................................... 99 13.1 Metric Space Baire Category Theorem .....................183 13.2 Locally Compact Hausdor Space Baire Category Theorem . 184 9 Hölder Spaces as Banach Spaces ===========================103 13.3Exercises...............................................190 9.1 Exercises...............................................107 14 Topological Vector Spaces =================================191 14.1BasicFacts.............................................191 Part III Topological Spaces: I 14.2 The Structure of Finite Dimensional Topological Vector Spaces 193 14.3MetrizableTopologicalVectorSpaces......................195 10 Topological Space Basics ===================================113 10.1ConstructingTopologiesandCheckingContinuity...........114 10.2ProductSpacesI ........................................120 Part V Calculus and Ordinary Dierential Equations in Banach 10.3Closureoperations.......................................123 Spaces 10.4CountabilityAxioms.....................................125 10.5Connectedness..........................................127 15 Ordinary Dierential Equations in a Banach Space ========203 10.6Exercises...............................................130 15.1Examples...............................................203 10.6.1GeneralTopologicalSpaceProblems.................130 15.2 Uniqueness Theorem and Continuous Dependence on Initial 10.6.2ConnectednessProblems...........................131 Data...................................................205 10.6.3MetricSpacesasTopologicalSpaces.................132 15.3LocalExistence(Non-LinearODE)........................207 15.4GlobalProperties........................................210 15.5 Semi-Group Properties of time independent flows............216 15.6Exercises...............................................218 6Contents Contents 7 16 Banach Space Calculus ====================================221 20 Fubini’s Theorem ==========================================319 16.1 The Dierential.........................................221 20.1 Fubini-Tonelli’s Theorem and Product Measure .............320 16.2ProductandChainRules.................................222 20.1.1 Application: More proofst.8.27 of the Weierstrass 16.3PartialDerivatives.......................................225 Approximation Theorem 8.27.......................327 16.4SmoothDependenceofODE’sonInitialConditions .........226 20.2 Lebesgue measure on Rg .................................330 16.5HigherOrderDerivatives.................................229 20.3PolarCoordinatesandSurfaceMeasure....................333 16.6ContractionMappingPrinciple............................233 20.4Exercises...............................................337 16.7InverseandImplicitFunctionTheorems....................235 16.8MoreontheInverseFunctionTheorem.....................239 21 Lp -spaces ==================================================339 16.8.1 Alternate construction of j .........................243 21.1Jensen’sInequality ......................................343 16.9Applications............................................243 21.2ModesofConvergence ...................................346 16.10Exercises...............................................245 21.3 Completeness of Os —spaces..............................350 21.3.1Summary:........................................354 21.4ConverseofHölder’sInequality............................355 Part VI Lebesbgue Integration Theory 21.5 Uniform Integrability ....................................361 21.6Exercises...............................................367 17 Introduction: What are measures and why “measurable” sets ========================================================251 22 Approximation Theorems and Convolutions ===============369 17.1TheproblemwithLebesgue“measure”.....................252 22.1ConvolutionandYoung’sInequalities......................374 22.1.1SmoothPartitionsofUnity.........................383 18 Measurability ==============================================257 22.2Exercises...............................................384 18.1 Algebras and —Algebras................................257 18.2MeasurableFunctions....................................262 18.2.1Moregeneralpointwiselimits.......................269 Part VII Construction of Measures 18.3 —FunctionAlgebras ...................................269 18.4 Product —Algebras....................................276 23 Uniqueness and Regulrity of Measures =====================389 18.4.1FactoringofMeasurableMaps ......................279 23.1 Monotone Class and — Theorems.......................389 18.5Exercises...............................................279 23.1.1Someotherproofsofpreviouslyprovedtheorems......392 23.2RegularityofMeasures...................................394 19 Measures and Integration ==================================281 19.1ExampleofMeasures ....................................284 24 Daniell — Stone Construction of Measures =================401 19.2 Integrals of Simple functions ..............................286 24.1 Finitely Additive Measures and Associated Integrals .........401 19.3 Integrals of positive functions .............................287 24.1.1 Integrals associated to finitelyadditivemeasures ......403 19.4 Integrals of Complex Valued Functions .....................295 24.2TheDaniell-StoneConstructionTheorem...................406 19.5 Measurability on Complete Measure Spaces .................303 24.3ExtensionsofpremeasurestomeasuresI....................410 19.6 Comparison of the Lebesgue and the Riemann Integral .......304 24.4RieszRepresentationTheorem............................414