An Introduction to Banach Space Theory (Graduate Texts in by Robert E. Megginson

By Robert E. Megginson

Getting ready scholars for additional research of either the classical works and present learn, this can be an obtainable textual content for college students who've had a path in actual and complicated research and comprehend the fundamental homes of L p areas. it really is sprinkled liberally with examples, ancient notes, citations, and unique resources, 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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21. 30 Let K be a compact Hausdorff space and let X be a normed space. 4, the collection of all continuous functions from K into X is a vector space when functions are added and multiplied by scalars in the usual way. :f: 0; 1f K 0 = 0. The resulting normed space is denoted by C(K, X). (a) Show that ll·lloo is in fact a norm on C(K, X). (b) Show that if X is a Banach space, then so is C(K,X). En x~ be a formal series in a Banach space. (a) Prove that if the series is absolutely convergent, then it is unconditionally convergent.

Xri- 1 }, a finite-dimensional subspace of X. The induction will be complete once a member Xn of Sx is found such that d(xn, Y) = 1, for then x 1 , ... xn- Xmil ~ 1 if m < n. Fix an element z of X\ Y and lett= d(z, Y), a positive number since Y is closed. A subspace of a vector space remains unchanged when multiplied by a nonzero scalar or translated by, one of its own elements, which in particular implies that t- 1Y = Y. It follows that 1 = t- 1d(z, Y) = d(t- 1z, t- 1Y) = d(t- 1z, Y). lYi- t- 1 zll-+ 1.

Thus, the ball Bx is·not convex. 24 1. 36 Suppose that X is the vector space underlying t 1 , but equipped with the leo norm. Show that {(an) : (an) E X, Lnlanl ~ 1} is a closed, convex, absorbing subset of X whose interior is empty. 37 Suppose that a vector space X has a nonempty subset B that is convex, balanced, and has this strong absorbing property: For every nonzero x in X, there is a positive S:z: such that x E tB if t ~ s, and x rl. tB if 0 ~ t < S:z:. Show that there is a norm II·IIB on X for which B is the closed unit ball.

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