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Folland: RealAnalysis, Chapter 2 S´ebastien Picard Problem2.3 If {fn} is a sequence of measurable functions on X, then {x : limfn(x) exists} is a measurable set. Solution: Deﬁne h = limsupfn, g = liminffn. By Proposition 2.7, h,g are measurable. Let E∞ = \∞ n=1 g−1(n,∞)∩ h−1(n,∞) E−∞ = \∞ n=1 g−1(−∞,−n) ∩h−1(−∞,−n).

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Folland: RealAnalysis, Chapter 5 S´ebastien Picard Problem5.7 Let Xbe a Banach space. a. If T∈ L(X,X) and ||I−T|| <1 where Iis the identity operator, then T is invertible; in fact, the series P∞ 0 (I−T)n converges in L(X,X) to T−1. b. If T ∈ L(X,X) is invertible and ||S− T|| <||T−1||−1, then Sis invertible. Thus the set of invertible operators in open in L(X,X). Solution: (a ...

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A Guide to AdvancedReal Analysis(MAA Guides #2), Gerald B. Folland MAA Service Center P.O. Box 91112 Washington,DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 “bevbook” — 2010/12/8 — 16:35 — page vii — #7 Preface The term “real analysis” refers, in the ﬁrst place, to the classical theory of functions of one and several real variables: limits and continuity, differen-tiation ...

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