Real Analysis Part II.pdf

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1、RealAnalysis:PartIIWilliamG.FarisJune3,2004iiContents1Functionspaces11.1Spacesofcontinuousfunctions...................11.2Pseudometricsandseminorms....................21.3Lpspaces...............................21.4DensesubspacesofLp........................51.5ThequotientspaceLp.............

2、...........61.6DualityofLpspaces.........................71.7Orliczspaces.............................81.8Problems...............................102Hilbertspace132.1Innerproducts............................132.2Closedsubspaces...........................162.3Theprojectiontheorem.............

3、..........172.4TheRiesz-Frechettheorem.....................182.5Bases.................................192.6SeparableHilbertspaces.......................212.7Problems...............................233Dierentiation273.1TheLebesguedecomposition....................273.2TheRadon-Nikodymtheorem

4、....................283.3Absolutelycontinuousincreasingfunctions.............293.4Problems...............................314ConditionalExpectation334.1Hilbertspaceideasinprobability..................334.2Elementarynotionsofconditionalexpectation...........354.3TheL2theoryofconditionalexpect

5、ation.............354.4TheL1theoryofconditionalexpectation.............374.5Problems...............................39iiiivCONTENTS5Fourierseries415.1Periodicfunctions..........................415.2Convolution..............................425.3Approximatedeltafunctions.....................42

6、5.4Summability.............................435.5L2convergence............................445.6C(T)convergence...........................465.7Pointwiseconvergence........................485.8Problems...............................496Fouriertransforms516.1Fourieranalysis......................

7、......516.2L1theory...............................526.3L2theory...............................536.4Absoluteconvergence.........................546.5Fouriertransformpairs........................556.6Poissonsummationformula.....................566.7Problem