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Licensed Unlicensed Requires Authentication Published by De Gruyter 1972

1.6. Finite Dimensional Families in Locally Convex Spaces

From the book Nuclear Locally Convex Spaces

  • Albrecht Pietsch
https://doi.org/10.1515/9783112564103-020
© 2022 Walter de Gruyter GmbH, Berlin/Munich/Boston
From the book
Nuclear Locally Convex Spaces
Nuclear Locally Convex Spaces

Chapters in this book (73)

Frontmatter
Foreword to the First Edition
Foreword to the Second Edition
Contents
Chapter O. Foundations
0.1. Topological Spaces
0.2. Metric Spaces
0.3. Linear Spaces
0.4. Semi-Norms
0.5. Locally Convex Spaces
0.6. The Topological Dual of a Locally Convex Space
0.7. Special Locally Convex Spaces
0.8. Banach Spaces
0.9. Hilbert Spaces
0.10. Continuous Linear Mappings in Locally Convex Spaces
0.11. The Normed Spaces Associated 'with a Locally Convex Space
0.12. Radon Measures
Chapter 1. Summable Families
1.1. Summable Families of Numbers
1.2. Weakly Summable Families in Locally Convex Spaces
1.3. Summable Families in Locally Convex Spaces
1.4. Absolutely Summable Families in Locally Convex Spaces
1.5. Totally Summable Families in Locally Convex Spaces
1.6. Finite Dimensional Families in Locally Convex Spaces
Chapter 2. Absolutely Summing Mappings
2.1. Absolutely Summing Mappings in Locally Convex Spaces
2.2. Absolutely Summing Mappings in Normed Spaces
2.3. A Characterization of Absolutely Summing Mappings in Normed Spaces
2.4. A Special Absolutely Summing Mappings
2.5. Hilbert-Schmidt Mappings
Chapter 3. Nuclear Mappings
3.1. Nuclear Mappings in Normed Spaces
3.2. Quasinuclear Mappings in Normed Spaces
3.3. Products of Quasinuclear and Absolutely Summing Mappings in Normed Spaces
3.4. The Theorem of Dvoretzky and Rogers
Chapter 4. Nuclear Locally Convex Spaces
4.1. Definition of Nuclear Locally Convex Spaces
4.2. Summable Families in Nuclear Locally Convex Spaces
4.3. The Topological Dual of Nuclear Locally Convex Spaces
4.4. Properties of Nuclear Locally Convex Spaces
Chapter 5. Permanence Properties of Nuclearity
5.1. Subspaces and Quotient Spaces
5.2. Topological Products and Sums
5.3. Complete Hulls
5.4. Locally Convex Tensor Products
5.5. Spaces of Continuous Linear Mappings
Chapter 6. Examples of Nuclear Locally Convex Spaces
6.1. Sequence Spaces
6.2. Spaces of Infinitely Differentiable Functions
6.3. Spaces of Harmonic Functions
6.4. Spaces of Analytic Functions
Chapter 7. Locally Convex Tensor Products
Introduction
7.1. Definition of Locally Convex Tensor Products
7.2. Special Locally Convex Tensor Products
7.3. A Characterization of Nuclear Locally Convex Spaces
7.4. The Kernel Theorem
7.5. The Complete π-Tensor Product of Normed Spaces
Chapter 8. Operators of Type l1 and s
8.1. The Approximation Numbers of Continuous Linear Mappings in Normed Spaces
8.2. Mappings of Type P
8.3. The Approximation Numbers of Compact Mappings in Hilbert Spaces
8.4. Nuclear and Absolutely Summing Mappings
8.5. Mappings of Type s
8.6. A Characterization of Nuclear Locally Convex Spaces
Chapter 9. Diametral and Approximative Dimension
9.1. The Diameter of Bounded Subsets in Normed Spaces
9.2. The Diametral Dimension of Locally Convex Spaces
9.3. The Diametral Dimension of Power Series Spaces
9.4. The Diametral Dimension of Nuclear Locally Convex Spaces
9.5. A Characterization of Dual Nuclear Locally Convex Spaces
9.6. The £-Entropy of Bounded Subsets in Normed Spaces
9.7. The Approximative Dimension of Locally Convex Spaces
9.8. The Approximative Dimension of Nuclear Locally Convex Spaces
Chapter 10. Nuclear Locally Convex Spaces with Basis
Introduction
10.1. Locally Convex Spaces with Basis
10.2. Representation of Nuclear Locally Convex Spaces with Basis
10.3- Bases in Special Nuclear Localty Convex Spaces
Chapter 11. Universal Nuclear Locally Convex Spaces
11.1. Imbedding in the Product Space (ξ)1
11.2. Embedding in the Product Space (ξ)1
Bibliography
Index
Table of Symbols
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