A short course on Banach space theory / N. L. Carothers.

Format:

Book

Author/Creator:

Carothers, N. L., 1952-

Series:

London Mathematical Society student texts ; 64.

London Mathematical Society student texts ; 64

Language:

English

Subjects (All):

Banach spaces.

Physical Description:

xii, 184 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2005.

Summary:

This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course concentrates on three major topics: The elementary theory of Schauder bases, an introduction to L[subscript p] spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, the primary focus here is on the postwar renaissance of Banach space theory brought about by James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their elegant and insightful results are useful in many contemporary research endeavors such as harmonic analysis, the theory of frames and wavelets, signal processing, economics, and physics.

The book is intended for use as in an advanced topics course or seminar or for independent study. This volume offers a gentler introduction than can be found in the existing literature and even includes elementary exercises. In addition, the text presents references to expository articles and suggestions for further reading.

Contents:

1 Classical Banach Spaces 1

The Sequence Spaces l[subscript p] and c[subscript 0] 1

Finite-Dimensional Spaces 2

The L[subscript p] Spaces 3

The C(K) Spaces 4

Hilbert Space 6

"Neoclassical" Spaces 7

The Big Questions 7

Continuous Linear Operators 11

Finite-Dimensional Spaces 12

Continuous Linear Functionals 13

Adjoints 15

Projections 16

Quotients 17

A Curious Application 20

3 Bases in Banach Spaces 24

Schauder's Basis for C[0, 1] 28

The Haar System 30

4 Bases in Banach Spaces II 34

A Wealth of Basic Sequences 34

Disjointly Supported Sequences in L[subscript p] and l[subscript p] 35

Equivalent Bases 38

5 Bases in Banach Spaces III 44

Block Basic Sequences 44

Subspaces of l[subscript p] and c[subscript 0] 47

Complemented Subspaces of l[subscript p] and c[subscript 0] 49

6 Special Properties of c[subscript 0], l[subscript 1], and l[subscript infinity] 55

True Stories About l[subscript 1] 55

The Secret Life of l[subscript infinity] 60

Confessions of c[subscript 0] 63

7 Bases and Duality 67

8 L[subscript p] Spaces 73

Basic Inequalities 73

Convex Functions and Jensen's Inequality 74

A Test for Disjointness 77

Conditional Expectation 78

9 L[subscript p] Spaces II 85

The Rademacher Functions 85

Khinchine's Inequality 87

The Kadec-Pelczynski Theorem 91

10 L[subscript p] Spaces III 99

Unconditional Convergence 99

Orlicz's Theorem 101

11 Convexity 107

Strict Convexity 108

Nearest Points 112

Smoothness 113

Uniform Convexity 114

Clarkson's Inequalities 117

An Elementary Proof That L*[subscript p] = L[subscript q] 119

12 C(K) Spaces 124

The Cantor Set 124

Completely Regular Spaces 125

13 Weak Compactness in L[subscript 1] 136

14 The Dunford-Pettis Property 142

15 C(K) Spaces II 148

The Stone-Cech Compactification 148

Return to C(K) 153

16 C(K) Spaces III 156

The Stone-Cech Compactification of a Discrete Space 156

A Few Facts About [beta] N 157

"Topological" Measure Theory 158

The Dual of l[subscript infinity] 161

The Riesz Representation Theorem for C([beta] D) 162

Appendix Topology Review 166

Separation 166

Locally Compact Hausdorff Spaces 167

Weak Topologies 169

Product Spaces 170

Nets 171.

Notes:

Includes bibliographical references (pages 173-179) and index.

ISBN:

0521842832

0521603722

OCLC:

54544081

Online:

Publisher description