Classical real analysis by Daniel Waterman (ed.)

By Daniel Waterman (ed.)

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The Banach space eo(N, C) is denoted eo(N). A sequence {Xn}::"~l is a Schauder or S-basis for a topological vector space E iff for all x in E there is in C a unique sequence {an}::"~l such that In anXn = x (convergence of the infinite series with respect to the topology of E). , according as the topology under consideration is norm-induced, (T(E, E*), etc. :l is partially ordered by inclusion. A set {xY}YEr is a basis for a topological vector space E iff for all x in E there is in C a unique set {aY}YEr such that the net I YE8 ayXy ~ x; I y ayXy denotes the limit of the net.

Problems 379-391 51 379. Show that if E is a Banach space and Me E* then (M1Y- is the weak* closure of span(M). 380. Show that if M is a convex subset of a Banach space X and M is norm-closed then M is also weakly closed. Show also that if M is convex then its norm-closure and weak closure are the same. 381. Show that if M is a norm-closed subspace of the Banach space X then (M1-)1- = M. 382. Give an example of a Banach space X and a norm-closed subspace M of X* such that (M1-)1- ~M. 383. Let E be a normed vector space and let F be a finite-dimensional subspace.

As n~CO then fE L 1(1R, A) and fn ~ fin L 1(1R, A) as n ~ co iff i) for each positive a there is a Lebesgue measurable set Aa such that A(Aa) < co and supn JRIAa Ifn (x)1 dx < a, and ii) limA(Bl-+O supn jB Ifn (x)1 dx = O. 299. 3 ... 2 n representingf: x ~ (1- X)-1/2 in (-1, 1) converge in Ll«-1, 1), A) to f. 300. e. )-313 n ~ 00, and II 1(lgn (x)1) dx ~ M n ~ 00. < 00, n in N. Show II Ign (x) - g (x)1 dx ~ 0 as 301. }. ). 302. ). 303. ) and let S be {x: x lim n .... oo II Icos Trl(x )In dx = J..

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