Functional Analysis: Introduction to Further Topics in by Elias M. Stein

By Elias M. Stein

Paperback version released in Asia. This publication is totally in English with the exact same contents because the unique variation.

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By Elias M. Stein

Paperback version released in Asia. This publication is totally in English with the exact same contents because the unique variation.

Show description

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Additional resources for Functional Analysis: Introduction to Further Topics in Analysis: Bk. 4

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In the end, we find that for n b N, |( 蜢 n)(f)| 蠄 f . This proves that 蜢 n 薔 0, as desired. In general, given a Banach space , it is interesting and very useful to be able to describe its dual *. This problem has an essentially complete answer in the case of the Lp spaces introduced before. 4 The dual space of Lp when 1 蠄 p ` 蜴 Suppose that 1 蠄 p 蠄 蜴 and q is the conjugate exponent of p, that is, 1/p + 1/q = 1. The key observation to make is the following: HɆlder舗s inequality shows that every function g Lq gives rise to a bounded linear functional on Lp by and that 蠄 g Lq.

First, if q = 1 and p = 蜴, we may take f(x) = sign g(x). Then, we have f L蜴 = 1, and clearly, fg = gL1. ) Then, if we take denotes the characteristic function of the set E, we see that f L1 = 1, and also This completes the proof of part (i). To prove (ii) we recall3 that we can find a sequence {gn} of simple functions so that |gn(x)| 蠄 |g(x)| while gn(x) 薔 g(x) for each x. When p b 1 (so q ` 蜴), we take As before, fnLp = 1. However and this does not exceed M. By Fatou舗s lemma it follows that Mq, so g Lq with gLq 蠄 M.

It is also exemplified by some of the specific questions that arose initially in the previous volumes and that are taken up again here: namely, the Dirichlet problem, ultimately treated by Brownian motion; the Radon transform, with its connection to Besicovitch sets; nowhere differentiable functions; and some problems in number theory, now formulated as distributions of lattice points. We hope that this choice of material will not only provide a broader view of analysis, but will also inspire the reader to pursue the further study of this subject.

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