Introduction to Banach Spaces and their Geometry

Name: Introduction to Banach Spaces and their Geometry
Brand: Elsevier
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Elsevier (Verlag)

Erschienen am 10. Oktober 2011

307 Seiten

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Inhalt

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

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