By N. L. Carothers

This brief path on classical Banach area conception is a traditional follow-up to a primary direction on practical research. the themes coated have confirmed worthy in lots of modern study 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 sophisticated subject matters direction or seminar, or for autonomous examine. It deals a extra ordinary creation than are available within the current literature and comprises references to expository articles and proposals for additional interpreting.

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De ne xn = e1 + + en in c0. Is (xn) a basis for c0? What is (xn) in this case? Is (xn) a basis for `1? 9. Prove that a normed space X is separable if and only if there is a sequence (xn) in X such that span(xn ) is dense in X . 10. Let X be a separable normed linear space. If E is any closed subspace of X , show that there is a sequence of norm-one functionals (fn) in X Tsuch that d(x; E ) = supn jfn (x)j for all x 2 X . Conclude that E = 1n=1 ker fn . ] 38 CHAPTER 3. BASES IN BANACH SPACES Chapter 4 Bases in Banach Spaces II We'll stick to the same notation throughout, with just a few exceptions.

That is, the coordinate functionals all have norm at most 2K . In particular, we always have 1 sup ja j 2K n n 1 X n=1 1 X anxn n=1 janj: Now, on with the proof. . 3) and hence, (yn) is a basic sequence equivalent to (xn). P P In other words, we've shown that the map T ( n anxn ) = n anyn is an isomorphism between xn ] and yn ]. Note that (4:3) gives kT k 1+2K < 2 and kT 1k (1 2K ) 1. To prove (ii), we next note that any nontrivial projection P has kP k 1, and hence the condition 8K kP k < 1 implies, at the very least, that 4K < 1.

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. 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.

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

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