Download An Introduction to Banach Space Theory by Robert E. Megginson PDF

By Robert E. Megginson

Getting ready scholars for additional research of either the classical works and present learn, this can be an available textual content for college students who've had a direction in actual and complicated research and comprehend the fundamental houses of L p areas. it's sprinkled liberally with examples, ancient notes, citations, and unique assets, and over 450 routines offer perform within the use of the consequences built within the textual content via supplementary examples and counterexamples.

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Of sets that are open, closed, or compact now follows, since homeomorphisms preserve these properties. Finally, if A and U are subsets of X and U is open, then A+ U = U{ a+ U: a E A}, a union of open sets. 4 Corollary. Let S be a topological space and X a normed space. + g)(s) = f(s) + g(s) and (a· f)(s) =a· f(s) for each sinS. PROOF. -. (f(s),g(s)) ~ (! -. -. af(s) from S into X are both continuous, that is, that f + g and af are continuous. The verification of the vector space axioms is then easy.

It follows that li~(o:Pk)X1 + · · · + a:~k)Xn) = 0:1X1 + · · · + O:nXn and that o:1x1 + · · · + O:nXn E Sw. Therefore every sequence in Sw has a subsequence converging to a member of Sw, so Sw is compact. Since I- 1 E L(W,X), the function w ~---+ III- 1 wll from W into IRis continuous and so attains a positive minimum on the compact set Sw. If I were not bounded, then there would be a sequence ( Zj) in B x such that I I Zj I ~ j for each j. Letting w1 = I I z11- 1 I z1 for each j yields a sequence (Wj) in Sw such that III- 1 wjll = 1Izil- 1 llzill-+ 0 as j-+ oo, a contradiction.

PROOF. If X is finite-dimensional, then it has the Heine-Borel property, so the closed bounded set Sx is compact. If X is infinite-dimensional, then the lemma produces a sequence in S x with no convergent subsequence, so Sx is not compact and X lacks the Heine-Borel property. 6. 34 1. 41 Suppose that Y is the vector space of finitely nonzero sequences, equipped with the f1 norm. Show that B(f1, Y) is not a Banach space. 42 Defined: IRxlR ->IR by the formula d(x, y) = ltan- 1(x)-tan- 1(y)l, where tan- 1 denotes the usual single-valued inverse tangent function from lR onto (-7r/2,7r/2).

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