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 |
I. Rieszโ{128}{153}s Lemma and Compactness in Banach Spaces. Isomorphic classification of finite dimensional Banach spaces โ{128}ฆ Rieszโ{128}{153}s lemma โ{128}ฆ finite dimensionality and compactness of balls โ{128}ฆ exercises โ{128}ฆ Kottmanโ{128}{153}s separation theorem โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- II. The Weak and Weak* Topologies: an Introduction. Definition of weak topology โ{128}ฆ non-metrizability of weak topology in infinite dimensional Banach spaces โ{128}ฆ Mazurโ{128}{153}s theorem on closure of convex sets โ{128}ฆ weakly continuous functional coincide with norm continuous functionals โ{128}ฆ the weak* topology โ{128}ฆ Goldstineโ{128}{153}s theorem โ{128}ฆ Alaogluโ{128}{153}s theorem โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- III. The Eberlein-ล mulian Theorem. Weak compactness of closed unit ball is equivalent to reflexivity โ{128}ฆ the Eberlein-ล mulian theorem โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- IV. The Orlicz-Pettis Theorem. Pettisโ{128}{153}s measurability theorem โ{128}ฆ the Bochner integral โ{128}ฆ the equivalence of weak subseries convergence with norm subseries convergence โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- V. Basic Sequences. Definition of Schauder basis โ{128}ฆ basic sequences โ{128}ฆ criteria for basic sequences โ{128}ฆ Mazurโ{128}{153}s technique for constructing basic sequences โ{128}ฆ Pelczynskiโ{128}{153}s proof of the Eberlein-ล mulian theorem โ{128}ฆ the Bessaga-Pelczynski selection principleโ{128}ฆ Banach spaces containing co โ{128}ฆ weakly unconditionally Cauchy series โ{128}ฆ co in dual spaces โ{128}ฆ basic sequences spanning complemented subspaces โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- VI. The Dvoretsky-Rogers Theorem. Absolutely P-summing operators โ{128}ฆ the Grothendieck-Pietsch domination theorem โ{128}ฆ the Dvoretsky-Rogers theorem โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- VII. The Classical Banach Spaces. Weak and pointwise convergence of se-quences in C(?) โ{128}ฆ Grothendieckโ{128}{153}s characterization of weak convergence โ{128}ฆ Baireโ{128}{153}s characterization of functions of the first Baire class โ{128}ฆ special features of co, l1l? โ{128}ฆ injectivity of l? โ{128}ฆ separable injectivity of co โ{128}ฆ projectivity of l1 โ{128}ฆ l1 is primary โ{128}ฆ Pelczynskiโ{128}{153}s decomposition method โ{128}ฆ the dual of l? โ{128}ฆ the Nikodym-Grothendieck boundedness theorem โ{128}ฆ Rosenthalโ{128}{153}s lemma โ{128}ฆ Phillipsโ{128}{153}s lemma โ{128}ฆ Schurโ{128}{153}s theorem โ{128}ฆ the Orlicz-Pettis theorem (again)โ{128}ฆ weak compactness in ca(?) and L1 (?) โ{128}ฆ the Vitali-Hahn-Saks theorem โ{128}ฆ the Dunford-Pettis theorem โ{128}ฆ weak sequential completeness of ca(?) and L1(?) โ{128}ฆ the Kadec-Pelczynski theorem โ{128}ฆ the Grothendieck-Dieudonne weak compactness criteria in rca โ{128}ฆ weak* convergent sequences in l?* are weakly convergent โ{128}ฆ Khintchineโ{128}{153}s Inequalities โ{128}ฆ Orliczโ{128}{153}s theorem โ{128}ฆ unconditionally convergent series in Lp[0, 1], 1 ? p ? 2 โ{128}ฆ the Banach-Saks theorem โ{128}ฆ Szlenkโ{128}{153}s theorem โ{128}ฆ weakly null sequences in Lp [0, 1], 1 ? p ? 2, have subsequences with norm convergent arithmetic means โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- VIII. Weak Convergence and Unconditionally Convergent Series in Uniformly Convex Spaces. Modulus of convexity โ{128}ฆ monotonicity and convexity properties of modulus โ{128}ฆ Kadecโ{128}{153}s theorem on unconditionally convergent series in uniformly convex spaces โ{128}ฆ the Milman-Pettis theorem on reflexivity of uniformly convex spaces โ{128}ฆ Kakutaniโ{128}{153}s proof that uniformly convex spaces have the Banach-Saks property โ{128}ฆ the Gurarii-Gurarii theorem on lp estimates for basic sequences in uniformly convex spaces โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- IX. Extremal Tests for Weak Convergence of Sequences and Series. The Krein-Milman theorem โ{128}ฆ integral representations โ{128}ฆ Bauerโ{128}{153}s characterization of extreme points โ{128}ฆ Milmanโ{128}{153}s converse to the Krein-Milman theorem โ{128}ฆ the Choquet integral representation theorem โ{128}ฆ Rainwaterโ{128}{153}s theorem โ{128}ฆ the Super lemma โ{128}ฆ Namiokaโ{128}{153}s density theorems โ{128}ฆ points of weak*-norm continuity of identity map โ{128}ฆ the Bessaga-Pelczynski characterization of separable duals โ{128}ฆ Haydonโ{128}{153}s separable generation theorem โ{128}ฆ the remarkable renorming procedure of Fonf โ{128}ฆ Eltonโ{128}{153}s extremal characterization of spaces without co-subspaces โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- X. Grothendieckโ{128}{153}s Inequality and the Grothendieck-Lindenstrauss-Pelczynski Cycle of Ideas. Rietzโ{128}{153}s proof of Grothendieckโ{128}{153}s inequality โ{128}ฆ definition of ?p spaces โ{128}ฆ every operator from a ?1-space to a ?2-space is absolutely 1-summing โ{128}ฆ every operator from a L? space to ?1 space is absolutely 2-summing โ{128}ฆ c0, l1 and l2 have unique unconditional bases โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- An Intermission: Ramseyโ{128}{153}s Theorem. Mathematical sociology โ{128}ฆ completely Ramsey sets โ{128}ฆ Nash-Williamsโ{128}{153} theorem โ{128}ฆ the Galvin-Prikry theorem โ{128}ฆ sets with the Baire property โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- XI. Rosenthalโ{128}{153}s l1-theorem. Rademacher-like systems โ{128}ฆ trees โ{128}ฆ Rosenthalโ{128}{153}s I1-theorem โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- XII. The Josefson-Nissenzweig Theorem. Conditions insuring l1โ{128}{153}s presence in a space given its presence in the dual โ{128}ฆ existence of weak* null sequences of norm-one functionals โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- XIII. Banach Spaces with Weak*-Sequentially Compact Dual Balls. Separable Banach spaces have weak* sequentially compact dual balls โ{128}ฆ stability results โ{128}ฆ Grothendieckโ{128}{153}s approximation criteria for relative weak compactness โ{128}ฆ the Davis-Figiel-Johnson-Pelczynski scheme โ{128}ฆ Amir-Lindenstrauss theorem โ{128}ฆ subspaces of weakly compactly generated spaces have weak* sequentially compact dual balls โ{128}ฆ so do spaces each of whose separable subspaces have a separable dual, thanks to Hagler and Johnson โ{128}ฆ the Odell-Rosenthal characterization of separable spaces with weak* sequentially compact second dual balls โ{128}ฆ exercises โ{128}ฆ notes and remarks โ{128}ฆ bibliography. -- XIV. The Elton-Odell (l + ?)Separation Theorem. Jamesโ{128}{153}s co-distortion theorem โ{128}ฆ Johnsonโ{128}{153}s combinatorial designs for detecting coโ{128}{153}s presence โ{128}ฆ the Elton-Odell proof that each infinite dimensional Banach space contains a (l + ?)-separated sequence of norm-one elements โ{128}ฆ exercises โ{128}ฆ notes and remarks . . bibliography