Lecture_Notes_on_Analysis-Kuttler.pdf

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LectureNotes
Kuttler
October8,2006
2
Contents
IPreliminaryMaterial 9
1SetTheory 11
1.1BasicDe¯nitions............................. 11
1.2TheSchroderBernsteinTheorem.................... 14
1.3EquivalenceRelations.......................... 17
1.4PartiallyOrderedSets.......................... 18
2TheRiemannStieltjesIntegral 19
2.1UpperAndLowerRiemannStieltjesSums............... 19
2.2Exercises ................................. 23
2.3FunctionsOfRiemannIntegrableFunctions.............. 24
2.4PropertiesOfTheIntegral........................ 27
2.5FundamentalTheoremOfCalculus................... 31
2.6Exercises ................................. 35
3ImportantLinearAlgebra 37
3.1AlgebrainF n ............................... 39
3.2SubspacesSpansAndBases....................... 40
3.3AnApplicationToMatrices....................... 44
3.4TheMathematicalTheoryOfDeterminants.............. 46
3.5TheCayleyHamiltonTheorem..................... 59
3.6AnIdentityOfCauchy.......................... 60
3.7BlockMultiplicationOfMatrices.................... 61
3.8Shur'sTheorem.............................. 63
3.9TheRightPolarDecomposition..................... 69
3.10TheSpace L (F n ; F m ) .......................... 71
3.11TheOperatorNorm........................... 72
4TheFrechetDerivative 75
4.1 C 1 Functions............................... 78
4.2 C k Functions............................... 83
4.3MixedPartialDerivatives........................ 83
4.4ImplicitFunctionTheorem....................... 85
4.5MoreContinuousPartialDerivatives.................. 89
3
4 CONTENTS
IILectureNotesForMath641and642 91
5MetricSpacesAndGeneralTopologicalSpaces 93
5.1MetricSpace............................... 93
5.2CompactnessInMetricSpace...................... 95
5.3SomeApplicationsOfCompactness................... 98
5.4AscoliArzelaTheorem..........................100
5.5GeneralTopologicalSpaces.......................103
5.6ConnectedSets..............................109
5.7Exercises .................................112
6ApproximationTheorems 115
6.1TheBernsteinPolynomials .......................115
6.2StoneWeierstrassTheorem.......................117
6.2.1TheCaseOfCompactSets...................117
6.2.2TheCaseOfLocallyCompactSets...............120
6.2.3TheCaseOfComplexValuedFunctions............121
6.3Exercises .................................122
7AbstractMeasureAndIntegration 125
7.1 ¾ Algebras.................................125
7.2TheAbstractLebesgueIntegral.....................133
7.2.1PreliminaryObservations....................133
7.2.2De¯nitionOfTheLebesgueIntegralForNonnegativeMea-
surableFunctions.........................135
7.2.3TheLebesgueIntegralForNonnegativeSimpleFunctions..136
7.2.4SimpleFunctionsAndMeasurableFunctions.........139
7.2.5TheMonotoneConvergenceTheorem.............140
7.2.6OtherDe¯nitions.........................141
7.2.7Fatou'sLemma..........................142
7.2.8TheRighteousAlgebraicDesiresOfTheLebesgueIntegral .144
7.3TheSpace L 1 ...............................145
7.4VitaliConvergenceTheorem.......................151
7.5Exercises .................................153
8TheConstructionOfMeasures 157
8.1OuterMeasures..............................157
8.2Regularmeasures.............................163
8.3Urysohn'slemma.............................164
8.4PositiveLinearFunctionals.......................169
8.5OneDimensionalLebesgueMeasure..................179
8.6TheDistributionFunction........................179
8.7CompletionOfMeasures.........................181
8.8ProductMeasures ............................185
8.8.1GeneralTheory..........................185
8.8.2CompletionOfProductMeasureSpaces............189
CONTENTS
5
8.9DisturbingExamples...........................191
8.10Exercises .................................193
9LebesgueMeasure 197
9.1BasicProperties .............................197
9.2TheVitaliCoveringTheorem......................201
9.3TheVitaliCoveringTheorem(ElementaryVersion)..........203
9.4VitaliCoverings..............................206
9.5ChangeOfVariablesForLinearMaps.................209
9.6ChangeOfVariablesFor C 1 Functions.................213
9.7MappingsWhichAreNotOneToOne.................219
9.8LebesgueMeasureAndIteratedIntegrals ...............220
9.9SphericalCoordinatesInManyDimensions..............221
9.10TheBrouwerFixedPointTheorem...................224
9.11Exercises .................................228
10The L p Spaces 233
10.1BasicInequalitiesAndProperties....................233
10.2DensityConsiderations..........................241
10.3Separability................................243
10.4ContinuityOfTranslation........................245
10.5Molli¯ersAndDensityOfSmoothFunctions .............246
10.6Exercises .................................249
11BanachSpaces 253
11.1TheoremsBasedOnBaireCategory..................253
11.1.1BaireCategoryTheorem.....................253
11.1.2UniformBoundednessTheorem.................257
11.1.3OpenMappingTheorem.....................258
11.1.4ClosedGraphTheorem.....................260
11.2HahnBanachTheorem..........................262
11.3Exercises .................................270
12HilbertSpaces 275
12.1BasicTheory...............................275
12.2ApproximationsInHilbertSpace....................281
12.3OrthonormalSets.............................284
12.4FourierSeries,AnExample.......................286
12.5Exercises .................................288
13RepresentationTheorems 291
13.1RadonNikodymTheorem........................291
13.2VectorMeasures .............................297
13.3RepresentationTheoremsForTheDualSpaceOf L p .........304
13.4TheDualSpaceOf C ( X )........................312
13.5TheDualSpaceOf C 0 ( X )........................314

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