Geometry of Banach spaces: selected topics
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1975
|
| Series: | Lecture notes in mathematics
485 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 282 S. |
| ISBN: | 3540074023 0387074023 |
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Record in the Search Index
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| adam_text | Table of Contents Chapter One. Support Functionals. The Blshop-fhelps theorem. . . James characterization of weakly compact subsets of a Banach space. . . application of James theorem to criterion for the reflexivity of the space of continuous linear operators. Chapter Two. §1 . Convexity and Differentiability of Norms. 20 Smoothness and Cateaux Differentiability of the Norm: support 20 mappings. . . equivalence of smoothness at a point and Gateaux differen tiability of the norm at a point. . . strict convexity. . . duality of strictly convex and smooth Banach spaces. . . James orthogonality. . . criterion for right and left uniqueness of orthogonality relations. §2 . Frechet Differentiability and Local Uniform Convexity: 29 characterization of Frechet differentiability of norm at a point with norm-to-norm continuity of support mapping at the point. . .very smooth Banach spaces. . .density characters In very smooth Banach spaces. . .weak local uniform convexity and local uniform convexity. . .duality between local uniform convexity and Fréchet differentiability and between weak local uniform convexity and very smooth spaces. §3 . Convexity and Smoothness in Higher Dials: discussion of the 33 deterioration of convexity and smoothness of norms in higher duals of non-reflexive Banach spaces. §4 . Uniform Smoothness, Uniform Convexity and Their Duality: equivalence of uniform smoothness, uniformly Frechet differentiable norm and norm-to-norm uniform continuity of support maps. . . duality between uniformly smooth and uniformly convex Banach spaces. . uniformly convex spaces.
. reflexivity of 35
viti 55. structure. Convexity, normal structure and fixed point theorems; normal 38 . . non-expansive maps. . . weakly compact convex subsets having normal structure have the fixed point property with respect to the class of non-expansive maps. . . commutative families of non- expansive maps and existence of common fixed points. Chapter Three. Uniformly convex and uniformly smooth Banach spaces. §1 . The uniform convexity of the L^(u)-spaces (1 p ”). Unconditionally Convergent Series in Uniformly Convex §2 . 54 54 57 Banach Spaces. 5Յ. The Day-Nordlander theorem. 54. The modulus of smoothness and the divergence of series in 62 Banach spaces. §5. The moduli of convexity of the 56. Bases In uniformly convex and uniformly smooth Banach spaces. 68 73 spaces (the theorem of V. I. Curarli and N. I. Curarli). §7. The Banach-Saks property; Kakutani s theorem showing all 73 uniformly convex Banach spaces have the Banach-Saks property. . . spaces with Banach-Saks property are reflexive. Chapter Four. §1· The classical renorming theorems. Day s norm on cp (Г) : J. Rainwter s norm on cQ(r) is locally uniformly convex. 94 proof that Μ. Μ. Day s 94
IX §2 , General facts about renorming: criteria for existence of iqq equivalent strictly convex noris. . . Troyanski s sufficient condition for existence of an equivalent locally uniformly convex norm. . . criteria for equivalent norm on a dual space to, be a dual norm. §3 . Asplund s averaging technique: the method of E. Asplund 106 to average two norms, one of a given degree of convexity, the other with a dual norm of a given degree of convexity, to obtain a third (equivalent) norm with both features. . §4 The Kadec-Klee-Asplund renorming theorem: if X is a i13 Banach space with separable dual then X can be renormed so that X* is smooth and locally uniformly convex. §5 . Possible and impossible renormings of while strictly 120 convexlflable is not smoothable nor weakly locally uniformly convexfiable . . . if Г is an uncountable set then lœ(r) is not strictly convexifiable. Weakly compactly generated Banach spaces. Chapter Five: §1 . Fundamental Lemmas: the construction of long sequences of 128 12g continuous linear projections in weakly compactly generated spaces. §2 . Basic Results in WCG Banach spaces: existence of continuous 143 injection of WCG Banach space into Сд(Г). . . resulting renorming theorems. . . weakly compact subsets oí Banach spaces always live in Cq(Г)-ѕрасеѕ. . . Eberlein compacts. . . Q is Eberlein compact if and only if C(O) is WCG. . . the dual ball of a WCG Banach space is weak-star sequentially compact. . . separable subspaces of WCG spaces are contained in complemented separable subspaces. . . operators on Grothendieck spaces... discrete
generation of WCG Banach spaces. . . the Johnson-Lindenstrauss stability criteria for WCG Banach spaces. §3 . Rosenthal s Topological Characterization of Eberlein Compacts: Gro th endi eck s criteria for weak compactness in C(G). . . C(G) is a WCG 15θ
x Banach algebra if and only if it is a WCG Banach space. . . Ω is Eberlein compact if and only if Ω admits a sequence (Q^) of point-finite families of open-5 sets such that U Q is separating. U П Ո §4 The Factorization of Weakly Compact Linear Operators: . the ֊ 160 remarkable factorization theorem of W. J. Davis, I. Figiel, W. В. Johnson and A. Pełczyński with applications. §5 . Tre,lanski s theorem: every WCG Banach space has an equivalent 164 locally uniformly convex norm. . . reflexive Banach spaces always have an equivalent norm; It and Its dual norm are locally uniformly convex and Frechet differentiable. §6 Operators Attaining Their Norm, . The Bishop-Phelps Property: 167 the collection of continuous linear operators between Banach spaces which achieve their maximum norm on the weak-star closure in X** of a closed bounded convex set of X is dense in the space of all operators. . . every weakly compact, convex subset of a Banach space is the closed convex hull of its strongly exposed points. 57. The Frledland-John-Zizler Theorem: If X is a WCG Banach space 173 and Y is a closed linear subspace of X with an equivalent very smooth norm then Y is WCG. . . in a WCG Banäch space with equivalent Frechet differentiable norm all closed linear subspaces are WCG. §8 . A Theorem of w. B. Johnson and J, Lindenstrauss: if X is a 177 Banach space with WCG dual and X embeds in a WCG Banach space, then X is WCG. §9 . The John-Zlzler Renortning Theorem: if X and X* are WCG, then X 165 can be renormed in a locally uniformly convex, Fréchet differentiable manner where the dual norm on
X* is also locally uniformly convex and smooth. §10 Spaces: . Counterexamples to General Stability Results for WCG Banach a discussion and description of the Johnson-Llndenstrauss example of a non-WCG Banach space with WCG dual and of Rosenthal s 189
XI example of a non-WCG closed linear subspace of an L1(p)-space for finite μ. The Radon-Nlkodým Theorem for Vector Measures. Chapter Six: }1. The Bochner Integral : review of notions of strong measurability, 199 199 Pettis integrability and Bochner integrability. §2 .Dentability and RAeffel s Criteria : the notion of dentabilit; 203 Μ. A. Rieffel s characterization of differentiable measures. . . examples of non-differentiable vector measures. §3 . The Davis-Huff-Maynard-Phelps Theorem: equivalence of the Radon- Nikodym property with dentability of bounded sets. . . stability criteria for Radon-Nikodym property. . . renorming spaces with Radon-Nikodym property. §4 . The Dunford-Pettis Theorem: 221 the classical result of N. Dunford and B. J. Pettis to the effect that separable dual spaces possess the Radon-Nikodym property. . . dual subspaces of WCG spaces have the Radon-Nikodym property. . . the Dunford-Pettis-Phillips theorem on representability of weakly compact operators on Լլ. . . weakly compact subsets of Լա(Ա) μ-flnite, are separable. §5 . A Lindenstrauss Result: The Krein-Milman property. . . the 230 Radon-Nikodym property implies the Krein-Milman property. §6 . The Huff-Morris-Stegall Theorem: in dual spaces the Radon- Nikodym property, the Krein-Milman property and the imbeddability 233 of separable subspaces into separable duals are equivalent. . . further stability results for the Radon-Nikodym property. . . If X* possesses the Radon-Nikodym property then bounded sequences in X have weak Cauchy subsequences. §7 . Edgar s Theorem: a Choquet type theorem for
separable closed 246 bounded convex subsets of a Banach space with the Radon-Nikodym property. §8 . A Theorem of R. R. Phelps: a lemma of E. Bishop. . . Phelps 252 characterization of the Radon-Nikodym property as that of closed bounded convex sets being the closed convex hull of strongly exposed points.
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| isbn | 3540074023 0387074023 |
| language | English |
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| physical | XI, 282 S. |
| publishDate | 1975 |
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| series2 | Lecture notes in mathematics |
| spelling | Diestel, Joseph 1943- Verfasser (DE-588)108479773 aut Geometry of Banach spaces selected topics Joseph Diestel Berlin [u.a.] Springer 1975 XI, 282 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 485 Banach, Espaces de Banachruimten gtt Mesures vectorielles Banach spaces Vector-valued measures Geometrie (DE-588)4020236-7 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Konvexität (DE-588)4114284-6 gnd rswk-swf Banach-Raum (DE-588)4004402-6 gnd rswk-swf Banach-Raum (DE-588)4004402-6 s Geometrie (DE-588)4020236-7 s DE-604 Konvexität (DE-588)4114284-6 s Funktionalanalysis (DE-588)4018916-8 s Lecture notes in mathematics 485 (DE-604)BV000676446 485 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001280183&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Diestel, Joseph 1943- Geometry of Banach spaces selected topics Lecture notes in mathematics Banach, Espaces de Banachruimten gtt Mesures vectorielles Banach spaces Vector-valued measures Geometrie (DE-588)4020236-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd Konvexität (DE-588)4114284-6 gnd Banach-Raum (DE-588)4004402-6 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4018916-8 (DE-588)4114284-6 (DE-588)4004402-6 |
| title | Geometry of Banach spaces selected topics |
| title_auth | Geometry of Banach spaces selected topics |
| title_exact_search | Geometry of Banach spaces selected topics |
| title_full | Geometry of Banach spaces selected topics Joseph Diestel |
| title_fullStr | Geometry of Banach spaces selected topics Joseph Diestel |
| title_full_unstemmed | Geometry of Banach spaces selected topics Joseph Diestel |
| title_short | Geometry of Banach spaces |
| title_sort | geometry of banach spaces selected topics |
| title_sub | selected topics |
| topic | Banach, Espaces de Banachruimten gtt Mesures vectorielles Banach spaces Vector-valued measures Geometrie (DE-588)4020236-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd Konvexität (DE-588)4114284-6 gnd Banach-Raum (DE-588)4004402-6 gnd |
| topic_facet | Banach, Espaces de Banachruimten Mesures vectorielles Banach spaces Vector-valued measures Geometrie Funktionalanalysis Konvexität Banach-Raum |
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