A Short Course on Banach Space Theory by N. L. Carothers

By N. L. Carothers

This brief direction on classical Banach area thought is a common follow-up to a primary path on practical research. the subjects coated have confirmed important in lots of modern study arenas, similar to harmonic research, the speculation of frames and wavelets, sign processing, economics, and physics. The e-book is meant to be used in a sophisticated subject matters path or seminar, or for self reliant examine. It bargains a extra elementary creation than are available within the latest literature and comprises references to expository articles and recommendations for extra analyzing.

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We consider the case T : `r ! `p where 1 r < p < 1 (and leave the remaining cases as an exercise). Suppose that T is an isomorphism from a subspace of `r onto a subspace W of `p. Then there is a further subspace Z of W and an isomorphism S : `p ! Z . But then T 1S is an isomorphism from `p into `r , which is impossible. Complemented subspaces of `p and c0 In this section, we present Pelczynski's characterization of the complemented subspaces of `p and c0 103]. His proof is based on an elegant and mysterious decomposition method.

If (xn) is a basis for a Banach space X , under what circumstances is (xn=kxn k) also a basis? In other words, can we always assume that the basis vectors are norm one? 5. If (xn) is a basis for a Banach space X , under what circumstances can we renormalize so as to have kxn k = kxnk = 1 for all n? 6. Let (P fn) be a disjointly supported, norm-one sequence in Lp( ). Show P 1 1 that n=1 an fn converges in Lp( ) if and only if n=1 janjp < 1. What, if anything, is the analogue of this result when p = 1?

5. 7 (The Principle of Small Perturbations) Let (xn) be a basic sequence in a Banach space X , with corresponding coordinate functionals (xn). P 1 Suppose that (yn) is a sequence in X with n=1 kxnkkxn ynk = . (i) If < 1, then (yn) is a basic sequence equivalent to (xn). (ii) If xn ] is the range of a projection P : X ! X , and if kP k < 1, then yn ] is complemented in X . Hint: For (ii), show that the map A : X ! X de ned by Ax = x Px + 1 X n=1 xn (Px) yn satis es kI Ak < 1 and Axn = yn . The projection onto yn ] is then given by Q = APA 1.

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