Sequences and Series in Banach Spaces

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.

Inhalt
I. Riesz's Lemma and Compactness in Banach Spaces. Isomorphic classification of finite dimensional Banach spaces Riesz's lemma finite dimensionality and compactness of balls exercises Kottman's separation theorem notes and remarks bibliography..- II. The Weak and Weak Topologies: an Introduction. Definition of weak topology non-metrizability of weak topology in infinite dimensional Banach spaces Mazur's theorem on closure of convex sets weakly continuous functional coincide with norm continuous functionals the weak topology Goldstine's theorem Alaoglu's theorem exercises notes and remarks bibliography..- III. The Eberlein-mulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity the Eberlein-mulian theorem exercises notes and remarks bibliography..- IV. The Orlicz-Pettis Theorem. Pettis's measurability theorem the Bochner integral the equivalence of weak subseries convergence with norm subseries convergence exercises notes and remarks bibliography..- V. Basic Sequences. Definition of Schauder basis basic sequences criteria for basic sequences Mazur's technique for constructing basic sequences Pelczynski's proof of the Eberlein-mulian theorem the Bessaga-Pelczynski selection principle Banach spaces containing co weakly unconditionally Cauchy series co in dual spaces basic sequences spanning complemented subspaces exercises notes and remarks bibliography..- VI. The Dvoretsky-Rogers Theorem. Absolutely P-summing operators the Grothendieck-Pietsch domination theorem the Dvoretsky-Rogers theorem exercises notes and remarks bibliography..- VII. The Classical BanachSpaces. Weak and pointwise convergence of se-quences in C(?) Grothendieck's characterization of weak convergence Baire's characterization of functions of the first Baire class special features of co, l1l? injectivity of l? separable injectivity of co projectivity of l1 l1 is primary Pelczynski's decomposition method the dual of l? the Nikodym-Grothendieck boundedness theorem Rosenthal's lemma Phillips's lemma Schur's theorem the Orlicz-Pettis theorem (again) weak compactness in ca(?) and L1 (?) the Vitali-Hahn-Saks theorem the Dunford-Pettis theorem weak sequential completeness of ca(?) and L1(?) the Kadec-Pelczynski theorem the Grothendieck-Dieudonne weak compactness criteria in rca weak convergent sequences in l? are weakly convergent Khintchine's Inequalities Orlicz's theorem unconditionally convergent series in Lp[0, 1], 1 ? p ? 2 the Banach-Saks theorem Szlenk's theorem weakly null sequences in Lp [0, 1], 1 ? p ? 2, have subsequences with norm convergent arithmetic means exercises notes and remarks bibliography..- VIII. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces. Modulus of convexity monotonicity and convexity properties of modulus Kadec's theorem on unconditionally convergent series in uniformly convex spaces the Milman-Pettis theorem on reflexivity of uniformly convex spaces Kakutani's proof that uniformly convex spaces have the Banach-Saks property the Gurarii-Gurarii theorem on lp estimates for basic sequences in uniformly convex spaces exercises notes and remarks bibliography..- IX. Extremal Tests for Weak Convergence ofSequences and Series. The Krein-Milman theorem integral representations Bauer's characterization of extreme points Milman's converse to the Krein-Milman theorem the Choquet integral representation theorem Rainwater's theorem the Super lemma Namioka's density theorems points of weak-norm continuity of identity map the Bessaga-Pelczynski characterization of separable duals Haydon's separable generation theorem the remarkable renorming procedure of Fonf Elton's extremal characterization of spaces without co-subspaces exercises notes and remarks bibliography..- X. Grothendieck's Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas. Rietz's proof of Grothendieck's inequality definition of ?p spaces every operator from a ?1-space to a ?2-space is absolutely 1-summing every operator from a L? space to ?1 space is absolutely 2-summing c0, l1 and l2 have unique unconditional bases exercises notes and remarks bibliography..- An Intermission: Ramsey's Theorem. Mathematical sociology completely Ramsey sets Nash-Williams' theorem the Galvin-Prikry theorem sets with the Baire property notes and remarks bibliography..- XI. Rosenthal's l1-theorem. Rademacher-like systems trees Rosenthal's I1-theorem exercises notes and remarks bibliography..- XII. The Josefson-Nissenzweig Theorem. Conditions insuring l1's presence in a space given its presence in the dual existence of weak null sequences of norm-one functionals exercises notes and remarks bibliography..- XIII. Banach Spaces with Weak-Sequentially Compact Dual Balls. Separable Banach spaces have weak sequentially compact dual balls stability results Grothendieck's approximation criteria for relative weak compactness the Davis-Figiel-Johnson-Pelczynski scheme Amir-Lindenstrauss theorem subspaces of weakly compactly generated spaces have weak sequentially compact dual balls so do spaces each of whose separable subspaces have a separable dual, thanks to Hagler and Johnson the Odell-Rosenthal characterization of separable spaces with weak sequentially compact second dual balls exercises notes and remarks bibliography..- XIV. The Elton-Odell (l + ?)Separation Theorem. James's co-distortion theorem Johnson's combinatorial designs for detecting co's presence the Elton-Odell proof that each infinite dimensional Banach space contains a (l + ?)-separated sequence of norm-one elements exercises notes and remarks . . bibliography..