巴拿赫空间中的序列和级数 Sequences and Series in Banach Spaces

基本信息

Format Paperback | 263 pages

Dimensions 152 x 229 x 14.73mm | 418g

Publication date 30 Jan 2012

Publisher Springer-Verlag New York Inc.

Language English

Edition Statement Softcover reprint of the original 1st ed. 1984

ISBN10 1461297346

ISBN13 9781461297345

页面参数仅供参考，具体以实物为准

书籍简介

本卷对巴拿赫空间理论中出现的一些具有一般分析性质的自然问题给出了答案。我相信，这里介绍的结果有太多不为活跃的抽象分析家所了解，而这是不应该的。巴拿赫空间理论有很多东西可以提供给分析的实践者；不幸的是，激励该理论并使其许多惊人成就得以实现的一些一般原则是用该领域的技术术语表述的，从而使不愿意花大量时间和精力来破译这些术语的人无法接近它。考虑到这一点，我集中介绍了我认为是任何分析家都能欣赏、喜欢、甚至使用的巴拿赫空间的基本现象。所涉及的主题至少有一个严重的遗漏：美丽而强大的类型和定型理论。坦率地说，如果不把文本的长度增加至少75%，我就无法说出我想说的关于这个主题的内容。即使如此，这些话也不如建议你去找丰富的毛利·施瓦茨研讨会（Seminaire Maurey-Schwartz）的讲义来得有用，在那里可以追溯到该理论从概念上的发展。同样，Lindenstrauss和Tzafriri的珍贵书卷也介绍了许多类型和同位素的理论，对于那些真正对巴拿赫空间理论感兴趣的人来说是必须阅读的。

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.

目录

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-Smulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity ... the Eberlein-Smulian 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-Smulian 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 Banach Spaces. 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 of Sequences 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..

作者简介

Joseph Diestel

Department of Math Sciences

Kent State UniversitKent, U.S.A