Introduction to Banach Spaces and their Geometry
Elsevier (Publisher)
Published on 10. October 2011
307 pages
978-0-08-087179-0 (ISBN)
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Introduction to Banach Spaces and their Geometry
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Bernard Beauzamy
Introduction to Banach Spaces and Their Geometry
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01/1982
North-Holland
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Content
- Front Cover
- Introduction to Banach Spaces and their Geometry
- Copyright Page
- Table of Contents
- INTRODUCTION
- CHAPTER 0. NOTATIONS AND PRELIMINARIES
- PART 1. FUNCTIONAL ANALYSIS
- CHAPTER I. BAIRE'S PROPERTY AND ITS CONSEQUENCES
- CHAPTER II. INFINITE-DIMENSIONAL NORMED SPACES
- § 1. Hahn-Banach Theorem : Analytic form
- § 2. Weak topology on the dual E*
- § 3. The bidual E** and the weak topology on E
- § 4. The separation o f convex subsets : the geometric form o f Hahn-Banach Theorem
- § 5. Closed convex sets and bounded sets in the weak an in the strong topologies
- § 6. Weak and strong continuity - Transposition
- § 7. Duality between subspaces and quotients for weak and strong topologies
- CHAPTER III. REFLEXIVE BANACH SPACES
- SEPARABLE BANACH SPACES
- § 1. Reflexivity
- § 2. Separability
- § 3. The utilization of convergent sequences in order to define a topology
- Weak compactness
- PART 2. THE STRUCTURE OF SOME COMMON BANACH SPACES
- CHAPTER I. HILBERT SPACES
- CHAPTER II. SCHAUDER BASES IN BANACH SPACES
- § 1. Schauder bases
- § 2 . Unconditional Schauder bases
- CHAPTER III. COMPLEMENTED SUBSPACES IN BANACH SPACES
- CHAPTER IV . THE BANACH SPACES lp( 1 = p = + 8 ) AND Co
- § 1. Subspaces of lp( 1 = p = + 8) and Co
- § 2. Some particular properties of l1 and . l8
- CHAPTER V. EXTREME POINTS OF COMPACT CONVEX SETS AND THE BANACH SPACES l(K)
- § 1. Extreme points of compact convex sets
- § 2. The Banach spaces l(K)
- CHAPTER VI. THE BANACH SPACES Lp (O, A , µ) (1 = p = + 8) P
- §1. Subspaces of Lp (1 G p = + 8 ) P
- § 2. The space L1
- § 3. Banach valued functions
- µ
- PART 3. SOME METRIC PROPERTIES IN BANACH SPACES
- CHAPTER I. STRICT CONVEXITY AND SMOOTHNESS
- § 1. Strict convexity
- § 2. Smoothness and Gateaux - differentiability of the norm
- § 3. Duality between strict convexity and smoothness
- CHAPTER II. UNIFORM CONVEXITY AND UNIFORM SMOOTHNESS
- § 1. Uniform convexity
- § 2. Uniform smoothness and Uniform Fréchet Differentiability
- PART 4. THE GEOMETRY OF SUPER-REFLEXIVE BANACH SPACES
- CHAPTER I. FINITE REPRESENTABILITY AND SUPER-PROPERTIES OF BANACH SPACES
- § 1. Finite - Representability and Ultrapowers o f Banach spaces
- § 2. Super-properties of Banach spaces
- super - reflexive spaces
- § 3. The Tree Properties
- CHAPTER II. BASIC SEQUENCES IN SUPER-REFLEXIVE BANACH SPACES
- CHAPTER III. UNIFORMLY NON-SQUARE AND J-CONVEX BANACH SPACES
- § 1. Uniformly non-square Banach Spaces
- § 2. J-convex Banach Spaces
- CHAPTER IV. RENORMING SUPER-REFLEXIVE BANACH SPACES
- § 1. Banach-valued Martingales
- § 2. A sequence of norms on a super-reflexive space
- BIBLIOGRAPHY
- INDEX
