Summing and Nuclear Norms in Banach Space Theory

G. J. O. Jameson

doi:10.1017/CBO9780511569166

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Summing and Nuclear Norms in Banach Space Theory

Summing and Nuclear Norms in Banach Space Theory

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G. J. O. Jameson

Publisher:: Cambridge University Press
Online publication date:: October 2009
Print publication year:: 1987
Online ISBN:: 9780511569166
DOI:: https://doi.org/10.1017/CBO9780511569166

Subjects:: Mathematics (general), Mathematics, Abstract Analysis
Series:: London Mathematical Society Student Texts (8)

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This textbook is an introduction to the techniques of summing and nuclear norms. The author's aim is to present a clear and simple account of these ideas and to demonstrate the power of their application to a variety of Banach space questions. The style is expository and the only prerequisite is a beginner's course on Wormed linear spaces and a minimal knowledge of functional analysis. Thus, Dr Jameson is able to concentrate on important, central results and gives concrete and largely non-technical proofs, often supplying alternative proofs which both contribute something to the understanding. Final-year undergraduates and postgraduates in functional analysis will enjoy this introduction to the subject, and there are many examples and exercises throughout the text to help the reader and to demonstrate the range of application these techniques find. A list of references indicates the way for further reading.

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Summing and Nuclear Norms in Banach Space Theory

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Introduction
pp ix-xii

0 - Banach space background
pp 1-11

1 - Finite rank operators: trace and 1-nuclear norm
pp 12-23

2 - Finite sequences of elements: the quantities μ1, μ2
pp 24-30

3 - The summing norms
pp 31-45

4 - Other nuclear norms: duality with the summing norms
pp 46-53

5 - Pietsch's theorem and its applications
pp 54-63

6 - Averaging; type 2 and cotype 2 constants
pp 64-73

7 - More averaging: Khinchin's inequality and related results
pp 74-87

8 - Integral methods; Gaussian averaging
pp 88-96

9 - 2-dominated spaces
pp 97-103

10 - Grothendieck's inequality
pp 104-117

11 - The interpolation method for Grothendieck-type theorems
pp 118-122

12 - Results connected with the basis constant
pp 123-132

13 - Estimation of summing norms using a restricted number of elements
pp 133-138

14 - Piseer's theorem for π2, 1
pp 139-144

15 - Tensor products of operators
pp 145-149

16 - Trace duality revisited: integral norms
pp 150-155

17 - Applications of local reflexivity
pp 156-159

18 - Cone-summing norms
pp 160-169

References
pp 170-172

List of symbols
pp 173-173

Index
pp 174-174

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Summing and Nuclear Norms in Banach Space Theory

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