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Sequences and series in Banach spaces

Author: Joe Diestel
Publisher: New York : Springer-Verlag, ©1984.
Series: Graduate texts in mathematics, 92.
Edition/Format:   Print book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Diestel, Joseph, 1943-
Sequences and series in Banach spaces.
New York : Springer-Verlag, ©1984
(OCoLC)610176119
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Joe Diestel
ISBN: 0387908595 9780387908595 3540908595 9783540908593
OCLC Number: 9556781
Description: xi, 261 pages ; 25 cm.
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-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..
Series Title: Graduate texts in mathematics, 92.
Responsibility: Joseph Diestel.

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