Sequences and Series in Banach Spaces | J. Diestel | Taschenbuch | Graduate Texts in Mathematics | Paperback | xii | Englisch | 2012 | Springer US | EAN 9781461297345 - Taschenbuch

EAN (ISBN-13): 9781461297345
ISBN (ISBN-10): 1461297346
Gebundene Ausgabe
Taschenbuch
Erscheinungsjahr: 2012
Herausgeber: Springer-Verlag New York Inc.

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ISBN/EAN: 1461297346

ISBN - alternative Schreibweisen:
1-4612-9734-6, 978-1-4612-9734-5
Alternative Schreibweisen und verwandte Suchbegriffe:
Autor des Buches: diestel, schwartz frank, analysis banach spaces
Titel des Buches: graduate text mathematics, bana, sequences and series banach spaces

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Daten vom Verlag:

Autor/in: J. Diestel
Titel: Graduate Texts in Mathematics; Sequences and Series in Banach Spaces
Verlag: Springer; Springer US
263 Seiten
Erscheinungsjahr: 2012-01-30
New York; NY; US
Gedruckt / Hergestellt in Niederlande.
Sprache: Englisch
90,94 € (DE)
93,49 € (AT)
100,50 CHF (CH)
POD
XII, 263 p.

BC; Hardcover, Softcover / Mathematik/Analysis; Mathematische Analysis, allgemein; Verstehen; Banach; Banach Space; Banachscher Raum; Convexity; Sequences; Series; Spaces; choquet integral; compactness; differential equation; extrema; Analysis; BB

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..