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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.
Contents
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-Šmulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity ... the Eberlein-Šmulian 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-Šmulian 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..
