By N. L. Carothers

This brief direction on classical Banach house thought is a traditional follow-up to a primary direction on sensible research. the subjects lined have confirmed invaluable in lots of modern learn arenas, equivalent to harmonic research, the speculation of frames and wavelets, sign processing, economics, and physics. The publication is meant to be used in a sophisticated themes direction or seminar, or for self sustaining examine. It deals a extra basic creation than are available within the present literature and comprises references to expository articles and recommendations for extra examining.

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**Example text**

Is separability, for example, a three space property? 21. Let T : X ! Y be a bounded linear map from a normed space X onto a normed space Y . If M is a closed subspace of ker T , then there is a 26 22. 23. 24. 25. 26. 27. 28. 29. CHAPTER 2. PRELIMINARIES (unique) bounded linear map Te : X=M ! Y such that T = Teq, where q is the quotient map. Moreover, kTek = kT k. If X and Y are Banach spaces, and if T : X ! Y is a bounded linear map onto all of Y , then X= ker T is isomorphic to Y . Let X and Y be Banach spaces, and let T 2 B (X; Y ).

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?

Prove that T extends to a map T~ 2 B (X; Rn) with kT~k = kT k. Hint: Hahn-Banach] 3. Prove that every proper subspace M of a normed space X has empty interior. If M is a nite dimensional subspace of an in nite dimensional normed space X , conclude that M is nowhere dense in X . 4. Prove that B (X; Y ) is complete whenever Y is complete. 5. If Y is a dense linear subspace of a normed space X , show that Y = X , isometrically. 6. Prove Riesz's lemma: Given a closed subspace Y of a normed space X and an " > 0, there is a norm one vector x 2 X such that kx yk > 1 " for all y 2 Y .