Read e-book online An Introduction to Banach Space Theory PDF

By Robert E. Megginson

ISBN-10: 0387984313

ISBN-13: 9780387984315

Getting ready scholars for extra examine of either the classical works and present study, this can be an available textual content for college kids who've had a path in actual and intricate research and comprehend the fundamental houses of L p areas. it truly is sprinkled liberally with examples, ancient notes, citations, and unique resources, and over 450 workouts supply perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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6 Example. Suppose that X is fp, where 1 ~ p < oo, or Co· Let (en) be the sequence of standard unit vectors in X. 3 First Properties of Normed Spaces 19 (an) =En anen whenever (an) EX. This result does not extend to £00 , for if (an) is a member of i 00 whose terms do not tend to 0, then the sequence ( E:=l anen):=l is not Cauchy and therefore cannot converge. If a sequence (xn) in a normed space generates a convergent series, then it is often said that En Xn is a convergent series or that En Xn converges, even though En Xn is only the limit of the series and is not the series itself.

Then the following are equivalent. --t Y be a (a) The operator Tis continuous. (b) The operator T is continuous at 0. (c) The operator T is uniformly continuous on X. (d) The operator Tis bounded. (e) For some neighborhood U of 0 in X, the set T(U) is bounded in Y. (f) There is a nonnegative real number M such that liTxll 5 Mllxll for each x in X. (g) The quantity sup{ liTxll : x E Bx} is finite. PROOF. Suppose that Tis continuous at 0. x1-x2l1 < 6. Therefore Tis uniformly continuous on X, which proves that (b)=> (c).

9 shows: However, the different norms must all induce the same· topology. The first of the following two corollaries is obtained by observing that each normed space with finite dimension n must be isomorphic to Euclidean n-space, a Banach space. The second then follows from the first, since each complete subset of a metric space is closed in that space. 19 Corollary. Every finite-dimensional normed space is a Banach space. 20 Corollary. Every finite-dimensional subspace of a normed space is a closed subset of the space.

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An Introduction to Banach Space Theory by Robert E. Megginson

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