# Sequences and Series in Banach Spaces (Graduate Texts in Mathematics)

Language: English

Pages: 263

ISBN: 0387908595

Format: PDF / Kindle (mobi) / ePub

This volume presents answers to some natural questions of a general analytic character that arise in the theory of Banach spaces. I believe that altogether too many of the results presented herein are unknown to the active abstract analysts, and this is not as it should be. Banach space theory has much to offer the prac titioners of analysis; unfortunately, some of the general principles that motivate the theory and make accessible many of its stunning achievements are couched in the technical jargon of the area, thereby making it unapproachable to one unwilling to spend considerable time and effort in deciphering the jargon. With this in mind, I have concentrated on presenting what I believe are basic phenomena in Banach spaces that any analyst can appreciate, enjoy, and perhaps even use. The topics covered have at least one serious omission: the beautiful and powerful theory of type and cotype. To be quite frank, I could not say what I wanted to say about this subject without increasing the length of the text by at least 75 percent. Even then, the words would not have done as much good as the advice to seek out the rich Seminaire Maurey-Schwartz lecture notes, wherein the theory's development can be traced from its conception. Again, the treasured volumes of Lindenstrauss and Tzafriri also present much of the theory of type and cotype and are must reading for those really interested in Banach space theory.

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then the corresponding sequence (s = µxk) in ba(2N) satisfies limµn(A) = lim F, x,,(m) it n mEA = limxA(xn) = 0. n Phillips's lemma now tells us that 0=urn n Itn({J})j=Jim EIxn(J)I=limIIxnpI1 n if Okay, it is time for the Orlicz-Pettis theorem again-only this time we prove it in much the same way Orlicz and Pettis did it in the first place using Schur's theorem except that we use Phillips's lemma. PROOF OF THE ORLIcz-PErris THEOREM. As usual, there is some initial footwork making clear

Math., 25, 337-341 Veech, W. A. 1971. Short proof of Sobczyk's theorem. Proc. Amer. Math. Soc., 28, 627-628. Vitali, G. 1907. Sull'integrazione per serie. Rend. Circ. Mat. Palermo, 23, 137-155. Yosida, K and Hewitt, E. 1952. Finitely additive measures. Trans. Amer. Math. Sot-, 72, 46--66. Zippin, M. 1977. The separable extension problem. Israel J. Math., 26, 372-387. CHAPTER VIII Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces In this chapter, we prove

moreover, Ilx**II = IxKBx). So far the fact that BX is the for all f E closed unit ball of a Banach space has been exploited but sparingly. Look at x+andXX+ - IX z x , x_ = IXI2 x which are both nonnegative members of ba(26x). Of course, X - X+ - X ; for E 5 BX define pE - - E and consider µ(E) = X+(E)+ X`(pE). µ it a nonnegative member of ba(28x) for which x**x* - fBxx*(x) dp(x) holds for all x* E X*. Moreover, IIx**II = s(Bx). Using the integration with respect to finitely additive

Math., 50, 163-182. Van Duist, D. 1978. Reflexive and Superreflexive Banach Spaces. Amsterdam: Mathematical Centre Tracts, Volume 102. CHAPTER IX Extremal Tests for Weak Convergence of Sequences and Series This chapter has two theorems as foci. The first, due to the enigmatic Rainwater, states that for a bounded sequence in a Banach space X to converge weakly to the point x, it is necessary and sufficient that x*x = limnx*x hold for each extreme point x* of Br.. The second improves the

the following improvement of a result of Dubinsky, Pelczynski, and Rosenthal (1972). Theorem. If X is a Banach lattice, then X has cotype 2 if and only if X has the Orlicz property. Generally, it is so that spaces having cotype 2 have the Orlicz' property; however, it is not known if every Banach space with the Orlicz property has cotype 2. A. Pelczynski and P. Wojtaszczyk were studying absolutely summing operators from the disk algebra to I2 when they discovered their proof of what is