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

By N. L. Carothers

This brief direction on classical Banach area thought is a common follow-up to a primary path on sensible research. the subjects coated have confirmed helpful in lots of modern study 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 complicated themes direction or seminar, or for self sufficient examine. It deals a extra straight forward advent than are available within the present literature and contains references to expository articles and proposals for additional interpreting.

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6, is in turn equivalent to (en). 1=r : We've arrived at a contradiction: If this inequality were to hold for all scalars, then, in particular, we'd have n1=p C kT k n1=r for all n. Since p < r, this is impossible. Consequently, kTenkp ! 0. The proof in case T : c0 ! `p is virtually identical. With just a bit more work, we could improve this result to read: A bounded linear map T : `r ! `p, 1 p < r < 1, or T : c0 ! `p is compact . 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 .

Proof. The linear span of (fn ) is una ected if we replace each fn by fn =kfn kp. Thus, we may assume that each fn is norm one. Next, since the fk are disjointly supported we have: m X k =n ak fk p p Z X m = k =n p ak fk d = m X k =n jak jp Z jfk jp d = m X k =n jak jp; P for any P scalars (ak ). This tells us that 1n=1 anfn converges in Lp( 1 p ) if and only if n=1 janj < 1. ) Thus, the map en 7! fn extends to a linear isometry from `p onto fn ]. In particular, it follows that (fn) is a basic sequence in Lp( ).

BASES IN BANACH SPACES II Exercises Recall that awsequence (xn) in a normed space X converges weakly to x 2 X , written xn ! x, if f (xn) ! f (x) for every f 2 X . It's easy to see that xn w! x if and only if xn x w! 0. A sequence tending weakly to 0 is said to be weakly null . 1. Let (fn ) be a sequence of disjointly supported functions in Lp, 1 < p < 1. Prove that Pn fn converges in Lp if and only if Pn kfnkpp < 1. 2. Let (xn) be a disjointly supported, norm one sequence in c0. Prove that xn ] is isometric to c0 and complemented in c0 by a norm one projection.

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