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

By N. L. Carothers

This brief path on classical Banach house conception is a common follow-up to a primary path on useful research. the themes lined have confirmed beneficial in lots of modern learn arenas, akin to harmonic research, the speculation of frames and wavelets, sign processing, economics, and physics. The e-book is meant to be used in a complicated themes path or seminar, or for self sustaining examine. It deals a extra elementary creation than are available within the latest literature and contains references to expository articles and proposals for extra examining.

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While we could give an elementary proof, very similar to the one we used for Schauder's basis (see Exercise 3), it might be entertaining to give a slightly fancier proof. The proof we'll give borrows a small amount of terminology from probability. For each k = 0; 1; : : :, let Ak = f (i 1)=2k+1 ; i=2k+1 ) : i = 1; : : :; 2k+1 g. Claim: The linear span of h0; : : :; h2k+1 1 is the set of all step functions based on the intervals in Ak . That is, spanf h0; : : :; h2k+1 1 g = spanf I : I 2 Ak g: Why?

Ii) range T is closed in Y . (iii) There is a constant C < 1 such that inf fkx yk : y 2 ker T g C kTxk for all x 2 X . Let M be a closed subspace of a Banach space X and let q : X ! X=M o , where B o is the open be the quotient map. Prove that q(BXo ) = BX=M X o unit ball in X and BX=M is the open unit ball in X=M . We say that T 2 B (X; Y ) is a quotient map if T (BXo ) = BYo , where BXo denotes the open unit ball in X (respectively, Y ). Prove that T is a quotient map if and only if X= ker T is isometric to Y .

In particular, the hn are linearly independent. Note that the Schauder system is related to the Haar system by the formula R x n 1 fn (x) = 2 0 hn 1 (t) dt for n 1 (and f0 1). In fact, it was Schauder who proved that the Haar system forms a monotone basis for Lp 0; 1 ], for any 1 p < 1. While we could give an elementary proof, very similar to the one we used for Schauder's basis (see Exercise 3), it might be entertaining to give a slightly fancier proof. The proof we'll give borrows a small amount of terminology from probability.

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