Author	Diestel, Joseph. author
Title	Sequences and Series in Banach Spaces [electronic resource] / by Joseph Diestel
Imprint	New York, NY : Springer New York, 1984
Connect to	http://dx.doi.org/10.1007/978-1-4612-5200-9
Descript	XII, 263 p. online resource

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 pracยญ 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

I. Rieszโs Lemma and Compactness in Banach Spaces. Isomorphic classification of finite dimensional Banach spaces โฆ Rieszโs lemma โฆ finite dimensionality and compactness of balls โฆ exercises โฆ Kottmanโs separation theorem โฆ notes and remarks โฆ bibliography. -- II. The Weak and Weak* Topologies: an Introduction. Definition of weak topology โฆ non-metrizability of weak topology in infinite dimensional Banach spaces โฆ Mazurโs theorem on closure of convex sets โฆ weakly continuous functional coincide with norm continuous functionals โฆ the weak* topology โฆ Goldstineโs theorem โฆ Alaogluโs theorem โฆ exercises โฆ notes and remarks โฆ bibliography. -- III. The Eberlein-ล mulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity โฆ the Eberlein-ล mulian theorem โฆ exercises โฆ notes and remarks โฆ bibliography. -- IV. The Orlicz-Pettis Theorem. Pettisโs measurability theorem โฆ the Bochner integral โฆ the equivalence of weak subseries convergence with norm subseries convergence โฆ exercises โฆ notes and remarks โฆ bibliography. -- V. Basic Sequences. Definition of Schauder basis โฆ basic sequences โฆ criteria for basic sequences โฆ Mazurโs technique for constructing basic sequences โฆ Pelczynskiโs proof of the Eberlein-ล mulian theorem โฆ the Bessaga-Pelczynski selection principleโฆ Banach spaces containing co โฆ weakly unconditionally Cauchy series โฆ co in dual spaces โฆ basic sequences spanning complemented subspaces โฆ exercises โฆ notes and remarks โฆ bibliography. -- VI. The Dvoretsky-Rogers Theorem. Absolutely P-summing operators โฆ the Grothendieck-Pietsch domination theorem โฆ the Dvoretsky-Rogers theorem โฆ exercises โฆ notes and remarks โฆ bibliography. -- VII. The Classical Banach Spaces. Weak and pointwise convergence of se-quences in C(?) โฆ Grothendieckโs characterization of weak convergence โฆ Baireโs characterization of functions of the first Baire class โฆ special features of co, l1l? โฆ injectivity of l? โฆ separable injectivity of co โฆ projectivity of l1 โฆ l1 is primary โฆ Pelczynskiโs decomposition method โฆ the dual of l? โฆ the Nikodym-Grothendieck boundedness theorem โฆ Rosenthalโs lemma โฆ Phillipsโs lemma โฆ Schurโs theorem โฆ the Orlicz-Pettis theorem (again)โฆ weak compactness in ca(?) and L1 (?) โฆ the Vitali-Hahn-Saks theorem โฆ the Dunford-Pettis theorem โฆ weak sequential completeness of ca(?) and L1(?) โฆ the Kadec-Pelczynski theorem โฆ the Grothendieck-Dieudonne weak compactness criteria in rca โฆ weak* convergent sequences in l?* are weakly convergent โฆ Khintchineโs Inequalities โฆ Orliczโs theorem โฆ unconditionally convergent series in Lp[0, 1], 1 ? p ? 2 โฆ the Banach-Saks theorem โฆ Szlenkโs theorem โฆ weakly null sequences in Lp [0, 1], 1 ? p ? 2, have subsequences with norm convergent arithmetic means โฆ exercises โฆ notes and remarks โฆ bibliography. -- VIII. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces. Modulus of convexity โฆ monotonicity and convexity properties of modulus โฆ Kadecโs theorem on unconditionally convergent series in uniformly convex spaces โฆ the Milman-Pettis theorem on reflexivity of uniformly convex spaces โฆ Kakutaniโs proof that uniformly convex spaces have the Banach-Saks property โฆ the Gurarii-Gurarii theorem on lp estimates for basic sequences in uniformly convex spaces โฆ exercises โฆ notes and remarks โฆ bibliography. -- IX. Extremal Tests for Weak Convergence of Sequences and Series. The Krein-Milman theorem โฆ integral representations โฆ Bauerโs characterization of extreme points โฆ Milmanโs converse to the Krein-Milman theorem โฆ the Choquet integral representation theorem โฆ Rainwaterโs theorem โฆ the Super lemma โฆ Namiokaโs density theorems โฆ points of weak*-norm continuity of identity map โฆ the Bessaga-Pelczynski characterization of separable duals โฆ Haydonโs separable generation theorem โฆ the remarkable renorming procedure of Fonf โฆ Eltonโs extremal characterization of spaces without co-subspaces โฆ exercises โฆ notes and remarks โฆ bibliography. -- X. Grothendieckโs Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas. Rietzโs proof of Grothendieckโs inequality โฆ definition of ?p spaces โฆ every operator from a ?1-space to a ?2-space is absolutely 1-summing โฆ every operator from a L? space to ?1 space is absolutely 2-summing โฆ c0, l1 and l2 have unique unconditional bases โฆ exercises โฆ notes and remarks โฆ bibliography. -- An Intermission: Ramseyโs Theorem. Mathematical sociology โฆ completely Ramsey sets โฆ Nash-Williamsโ theorem โฆ the Galvin-Prikry theorem โฆ sets with the Baire property โฆ notes and remarks โฆ bibliography. -- XI. Rosenthalโs l1-theorem. Rademacher-like systems โฆ trees โฆ Rosenthalโs I1-theorem โฆ exercises โฆ notes and remarks โฆ bibliography. -- XII. The Josefson-Nissenzweig Theorem. Conditions insuring l1โs presence in a space given its presence in the dual โฆ existence of weak* null sequences of norm-one functionals โฆ exercises โฆ notes and remarks โฆ bibliography. -- XIII. Banach Spaces with Weak*-Sequentially Compact Dual Balls. Separable Banach spaces have weak* sequentially compact dual balls โฆ stability results โฆ Grothendieckโs approximation criteria for relative weak compactness โฆ the Davis-Figiel-Johnson-Pelczynski scheme โฆ Amir-Lindenstrauss theorem โฆ subspaces of weakly compactly generated spaces have weak* sequentially compact dual balls โฆ so do spaces each of whose separable subspaces have a separable dual, thanks to Hagler and Johnson โฆ the Odell-Rosenthal characterization of separable spaces with weak* sequentially compact second dual balls โฆ exercises โฆ notes and remarks โฆ bibliography. -- XIV. The Elton-Odell (l + ?)Separation Theorem. Jamesโs co-distortion theorem โฆ Johnsonโs combinatorial designs for detecting coโs presence โฆ the Elton-Odell proof that each infinite dimensional Banach space contains a (l + ?)-separated sequence of norm-one elements โฆ exercises โฆ notes and remarks . . bibliography

1. Mathematics
2. Mathematical analysis
3. Analysis (Mathematics)
4. Mathematics
5. Analysis