Royden_Real_Analysis_Solutions

RealAnalysisbyH.L.RoydenContents1SetTheory11.1Introduction............................................11.2Functions.............................................11.3Unions,intersectionsandcomplements.............................11.4Algebrasofsets..........................................21.5Theaxiomofchoiceandinnitedirectproducts.......................21.6Countablesets..........................................31.7Relationsandequivalences....................................31.8Partialorderingsandthemaximalprinciple..........................31.9Wellorderingandthecountableordinals............................32TheRealNumberSystem52.1Axiomsfortherealnumbers...................................52.2ThenaturalandrationalnumbersassubsetsofR.......................52.3Theextendedrealnumbers...................................52.4Sequencesofrealnumbers....................................52.5Openandclosedsetsofrealnumbers..............................72.6Continuousfunctions.......................................92.7Borelsets.............................................133LebesgueMeasure133.1Introduction............................................133.2Outermeasure..........................................143.3MeasurablesetsandLebesguemeasure.............................143.4Anonmeasurableset.......................................153.5Measurablefunctions.......................................153.6Littlewood'sthreeprinciples...................................174TheLebesgueIntegral184.1TheRiemannintegral......................................184.2TheLebesgueintegralofaboundedfunctionoverasetofnitemeasure..........184.3Theintegralofanonnegativefunction.............................194.4ThegeneralLebesgueintegral..................................194.5Convergenceinmeasure.....................................215DierentiationandIntegration225.1Dierentiationofmonotonefunctions..............................225.2Functionsofboundedvariation.................................235.3Dierentiationofanintegral...................................245.4Absolutecontinuity........................................245.5Convexfunctions.........................................266TheClassicalBanachSpaces276.1TheLpspaces...........................................276.2TheMinkowskiandHolderinequalities.............................276.3Convergenceandcompleteness.................................286.4ApproximationinLp.......................................296.5BoundedlinearfunctionalsontheLpspaces..........................307MetricSpaces307.1Introduction............................................307.2Openandclosedsets.......................................317.3Continuousfunctionsandhomeomorphisms..........................317.4Convergenceandcompleteness.................................327.5Uniformcontinuityanduniformity...............................337.6Subspaces.............................................357.7Compactmetricspaces......................................357.8Bairecategory..........................................367.9AbsoluteG's...........................................397.10TheAscoli-ArzelaTheorem...................................408TopologicalSpaces418.1Fundamentalnotions.......................................418.2Basesandcountability......................................438.3Theseparationaxiomsandcontinuousreal-valuedfunctions.................448.4Connectedness..........................................478.5Productsanddirectunionsoftopologicalspaces.......................488.6Topologicalanduniformproperties...............................508.7Nets................................................509CompactandLocallyCompactSpaces519.1Compactspaces..........................................519.2CountablecompactnessandtheBolzano-Weierstrassproperty................529.3Productsofcompactspaces...................................539.4Locallycompactspaces.....................................539.5-compactspaces.........................................569.6Paracompactspaces.......................................569.7Manifolds.............................................579.8TheStone-Cechcompactication................................579.9TheStone-WeierstrassTheorem.................................5810BanachSpaces60ii10.1Introduction............................................6010.2Linearoperators.........................................6110.3LinearfunctionalsandtheHahn-BanachTheorem......................6210.4TheClosedGraphTheorem...................................6310.5Topologicalvectorspaces....................................6410.6Weaktopologies.........................................6610.7Convexity.............................................6910.8Hilbertspace...........................................7111MeasureandIntegration7311.1Measurespaces..........................................7311.2Measurablefunctions.......................................7611.3Integration............................................7711.4Generalconvergencetheorems....