Download e-book for iPad: A Short Course on Banach Space Theory by N. L. Carothers

By N. L. Carothers

This brief path on classical Banach house idea is a traditional follow-up to a primary path on sensible research. the themes coated have confirmed important in lots of modern learn arenas, similar to harmonic research, the idea of frames and wavelets, sign processing, economics, and physics. The e-book is meant to be used in a sophisticated issues path or seminar, or for self sustaining examine. It deals a extra hassle-free advent than are available within the present literature and comprises references to expository articles and proposals for additional examining.

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That is, T maps bounded sets into compact sets. 8 actually shows something more: T fails to be an isomorphism on any in nite dimensional subspace of `r . " Or, in still other words, each in nite 54 CHAPTER 5. BASES IN BANACH SPACES III dimensional subspace of `r contains an isomorphic copy of `r and the restriction of T to this copy of `r cannot be an isomorphism. A bounded linear map T : X ! Y which fails to be an isomorphism on any subspace of X is said to be strictly singular . ) Thus, every map T : `r !

Show that (X Y )1 = (X Y )1, isometrically. 8. Prove that N can be written as the union of in nitely many pairwise disjoint in nite subsets. 9. Find a \natural" copy of (`p `p )p in Lp(R). 10. If (Xn ) is a sequence of Banach spaces, prove that (X1 X2 )p is a Banach space for any 1 p 1. 11. 10 requires that ((Y Z ) (Y Z ) )p (Z Z )p (Y Y )p. Verify this claim. 12. 10 in the case X = c0. Chapter 6 Special Properties of c0, `1, and `1 The spaces c0, `1 , `2, and `1 play very special roles in Banach space theory.

Show \by hand" that (xn) is not equivalent to (en) in `p, for 1 p < 1, but that (xn) is equivalent to (en) in c0. What can you say about the block basis yn = xn=n? 2. 6. 3. 7. 4. If T : c0 ! `p, 1 p < 1, is bounded and linear, show that kTenkp ! 0. 5. Show that every bounded linear map T : `r ! `p , 1 p < r < 1, or T : c0 ! `p, 1 p < 1, is compact. ] 6. Show that every bounded linear map T : c0 ! `r or T : `r ! c0, 1 r < 1, is strictly singular. 7. Show that (X Y )1 = (X Y )1, isometrically. 8.

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