[Article] Studies of Magnitudes in Star Clusters V. Further by Shapley H.

By Shapley H.

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Extra info for [Article] Studies of Magnitudes in Star Clusters V. Further Evidence of the Absence of Scattering of Light in Space

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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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