Nuclear Locally Convex Spaces

VI closely related to finite dimensional locally convex spaces than are normed spaces. In order to present a clear narrative I have omitted exact references to the literature for individual propositions. However, each chapter begins with a short introduction which also contains historical remarks. Deutsche Akademie der vVissenschaften zu Berlin Institut fur Reine Mathematik Albrecht Pietsch Foreword to the Second Edition Since the appearance of the first edition, some important advances have taken place in the theory of nuclear locally convex spaces. Firsts there is the Universality Theorem ofT. and Y. Komura which fully confirms a conjecture of Grothendieck. Also, of particular interest are some new existence theorems for bases in special nuclear locally convex spaces. Recently many authors have dealt with nuclear spaces of functions and distributions. Moreover, further classes of operators have been found which take the place of nuclear or absolutely summing operators in the theory of nuclear locally convex spaces. Unfortunately, there seem to be no new results on diametrai or approximative dimension and isomorphism of nuclear locally convex spaces. Since major changes have not been absolutely necessary I have restricted myself to minor additions. Only the tenth chapter has been substantially altered. Since the universality results no longer depend on the existence of a basis it was necessary to introduce an independent eleventh chapter on universal nuclear locally convex spaces. In the same chapter s-nuclear locally convex spaces are also briefly treated.

Table of Contents:
0. 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.- 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.- 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.- 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.- 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.- 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.- 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.- 7. Locally Convex Tensor Products.- 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 rc-Tensor Product of Normed Spaces.- 8. Operators of Type lp and s.- 8.1. The Approximation Numbers of Continuous Linear Mappings in Normed Spaces.- 8.2. Mappings of Type lp.- 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.- 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.- 10. Nuclear Locally Convex Spaces with Basis.- 10.1. Locally Convex Spaces with Basis.- 10.2. Representation of Nuclear Locally Convex Spaces with Basis.- 10.3. Bases in Special Nuclear Locally Convex Spaces.- 11. Universal Nuclear Locally Convex Spaces.- 11.1. Imbedding in the Product Space (?)1.- 11.2. Imbedding in the Product Space(?)1.- Table of Symbols.