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.

1), a totally bounded subset of X. Of course this says that given e > 0 each vector in ( fE f die : E E 2) can be approximated within e/2 by a vector in the totally bounded set (fEg d)A: E E 2 ), so ( JE f dµ : E E 2 ) is itself totally bounded. Now for the proof of the Orlicz-Pettis theorem. is weakly Let's imagine what could go wrong with the theorem. If subseries convergent (i.e., satisfies the hypotheses of the Orlicz-Pettis theo- rem) yet fails to be norm subseries convergent, it's because

functional p E C(Bx., weak*)* (i.e., regular Borel measures on BX. in its weak* topology) such that ffd'`=µ(f)s0

_ 0 for each x c- X. But this just says that IITxIIP s irI(T) f

that the sequence (fk ) satisfies the following: first, (fk. / II fk ID is a basic sequence equivalent to tie unit vector basis of 11, and second, tine closed linear span [fk.] of the f is complemented in L1[O,11 (and, of course, isomorphic to 11). Let g,, fk,, g, s fk., and g - g r. Of course, (g;; : n;-* 1) is relatively weakly compact in L1[0,1]; so with perhaps another turn at extracting subsequences, we may assume (g;') is weakly convergent. Now notice that we've located a sequence g in

precisely, we have the following theorem. Theorem 2. For any compact subset C in E, a point x of E belongs to the closed convex hull of C if and only if there exists a regular Borel probability measure µ on C whose barycenter (exists and) is x. 150 IX. Extremal Tests for Weak Convergence of Sequences and Series PROOF. If p is a regular Bore] probability measure on C and has x as its barycenter, then for any f E E * we know f(x)= f fdµ_supf(C)ssupf(coC). C Were x not in coC, there would be

dual. 2. Each nonempty closed bounded convex subset of X* is deniable. 3. Each nonempty closed bounded convex subset of X* has an extreme point. Theorem 10 was established by R. Haydon (1976), K. Musial (1978), and V. I. Rybakov (1977); our proof was inspired by Haydon's, but its execution differs at several crucial junctures. Subsequent to stumbling onto this variation in approach, E. Saab (1977) pointed out that he had used the same tactics to much greater advantage in deriving several