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A Short Course on Banach Space Theory : London Mathematical Society Student Texts - N. L. Carothers

A Short Course on Banach Space Theory

London Mathematical Society Student Texts

Paperback

Published: 27th April 2005
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This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course focuses on three major topics: The elementary theory of Schauder bases, an introduction to Lp spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, our primary interest is in 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 and deserve greater publicity. By way of prerequisites, the reader will need an elementary understanding of functional analysis and at least a passing familiarity with abstract measure theory. An introductory course in topology would also be helpful; however, the text includes a brief appendix on the topology needed for the course.

'This lively written text focuses on certain aspects of the (neo-) classical theory of Banach spaces as developed in the 1950s and 1960s and is intended as an introduction to the subject, e.g., for future Ph.D. students. ... This slim book is indeed very well suited to serve as an introduction to Banach spaces. Readers who have mastered it are well prepared to study more advanced texts such as P. Wojtaszczyk's Banach Spaces for Analysts (Cambridge University Press, second edition) or research papers.' Zentralblatt MATH '... a painstaking attention both to detail in the mathematics and to accessibility for the reader. ... You could base a good postgraduate course on it.' Bulletin of the London Mathematical Society

Prefacep. xi
Classical Banach Spacesp. 1
The Sequence Spaces l[subscript p] and c[subscript 0]p. 1
Finite-Dimensional Spacesp. 2
The L[subscript p] Spacesp. 3
The C(K) Spacesp. 4
Hilbert Spacep. 6
"Neoclassical" Spacesp. 7
The Big Questionsp. 7
Notes and Remarksp. 9
Exercisesp. 9
Preliminariesp. 11
Continuous Linear Operatorsp. 11
Finite-Dimensional Spacesp. 12
Continuous Linear Functionalsp. 13
Adjointsp. 15
Projectionsp. 16
Quotientsp. 17
A Curious Applicationp. 20
Notes and Remarksp. 20
Exercisesp. 20
Bases in Banach Spacesp. 24
Schauder's Basis for C[0, 1]p. 28
The Haar Systemp. 30
Notes and Remarksp. 32
Exercisesp. 33
Bases in Banach Spaces IIp. 34
A Wealth of Basic Sequencesp. 34
Disjointly Supported Sequences in L[subscript p] and l[subscript p]p. 35
Equivalent Basesp. 38
Notes and Remarksp. 41
Exercisesp. 42
Bases in Banach Spaces IIIp. 44
Block Basic Sequencesp. 44
Subspaces of l[subscript p] and c[subscript 0]p. 47
Complemented Subspaces of l[subscript p] and c[subscript 0]p. 49
Notes and Remarksp. 51
Exercisesp. 54
Special Properties of c[subscript 0], l[subscript 1], and l[subscript infinity]p. 55
True Stories About l[subscript 1]p. 55
The Secret Life of l[subscript infinity]p. 60
Confessions of c[subscript 0]p. 63
Notes and Remarksp. 65
Exercisesp. 65
Bases and Dualityp. 67
Notes and Remarksp. 71
Exercisesp. 72
L[subscript p] Spacesp. 73
Basic Inequalitiesp. 73
Convex Functions and Jensen's Inequalityp. 74
A Test for Disjointnessp. 77
Conditional Expectationp. 78
Notes and Remarksp. 82
Exercisesp. 83
L[subscript p] Spaces IIp. 85
The Rademacher Functionsp. 85
Khinchine's Inequalityp. 87
The Kadec-Pelczynski Theoremp. 91
Notes and Remarksp. 97
Exercisesp. 98
L[subscript p] Spaces IIIp. 99
Unconditional Convergencep. 99
Orlicz's Theoremp. 101
Notes and Remarksp. 106
Exercisesp. 106
Convexityp. 107
Strict Convexityp. 108
Nearest Pointsp. 112
Smoothnessp. 113
Uniform Convexityp. 114
Clarkson's Inequalitiesp. 117
An Elementary Proof That L*[subscript p] = L[subscript q]p. 119
Notes and Remarksp. 122
Exercisesp. 122
C(K) Spacesp. 124
The Cantor Setp. 124
Completely Regular Spacesp. 125
Notes and Remarksp. 134
Exercisesp. 134
Weak Compactness in L[subscript 1]p. 136
Notes and Remarksp. 141
Exercisesp. 141
The Dunford-Pettis Propertyp. 142
Notes and Remarksp. 146
Exercisesp. 147
C(K) Spaces IIp. 148
The Stone-Cech Compactificationp. 148
Return to C(K)p. 153
Notes and Remarksp. 155
Exercisesp. 155
C(K) Spaces IIIp. 156
The Stone-Cech Compactification of a Discrete Spacep. 156
A Few Facts About [beta] Np. 157
"Topological" Measure Theoryp. 158
The Dual of l[subscript infinity]p. 161
The Riesz Representation Theorem for C([beta] D)p. 162
Notes and Remarksp. 165
Exercisesp. 165
Topology Reviewp. 166
Separationp. 166
Locally Compact Hausdorff Spacesp. 167
Weak Topologiesp. 169
Product Spacesp. 170
Netsp. 171
Notes and Remarksp. 172
Exercisesp. 172
Referencesp. 173
Indexp. 181
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521603720
ISBN-10: 0521603722
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 184
Published: 27th April 2005
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.09  x 1.12
Weight (kg): 0.27