Functional Analysis

Name: Functional Analysis | Theory and Applications
Brand: Dover Publications
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Theory and Applications

R. E. Edwards(Autor*in)

Dover Publications (Verlag)

Erschienen am 25. Oktober 2012

1764 Seiten

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Inhalt

Cover
Title Page
Copyright Page
Preface
Contents
Chapter 0 Preliminaries on Set Theory and Topology
0.0 Foreword
0.1 Preliminaries from Set Theory
0.2 Preliminaries from General Topology
0.3 Uniform Spaces
0.4 Ascoli's Theorem
0.5 Brouwer's Fixed-point Theorem
Chapter 1 Vector Spaces and Topological Vector Spaces
1.0 Foreword
1.1 Vector Spaces
1.2 Vector Subspaces, Quotient Spaces, Product Spaces
1.3 Linear Maps and Linear Forms. Isomorphism. Graphs
1.4 Linear Dependence. Algebraic Bases. Algebraic Complements
1.5 Absorbent Sets. Balanced Sets. Convex Sets
1.6 Gauges and Seminorms
1.7 The Hahn-Banach Theorem in Analytic Form
1.8 Topological Vector Spaces
1.9 Continuity of Linear Maps. Isomorphisms and Homomorphisms of Topological Vector Spaces. Closed Linear Maps
1.10 Locally Convex Topological Vector Spaces
1.11 Weak Topologies on Dual Spaces
1.12 Inner Product Spaces, pre-Hilbert Spaces, Hilbert Spaces
Chapter 2 The Hahn-Banach Theorem
2.1 Foreword
2.2 Geometric Forms of the Hahn-Banach Theorem
2.3 The Extension Principle
2.4 The Approximation Principles
2.5 Supplements to, and Further Applications of, Theorem 2.4.1
2.6 Extension of Positive Linear Forms
2.7 Some Results with Applications to the Theory of Games
Chapter 3 Fixed-Point Theorems
3.0 Foreword
3.1 Fixed-point Theorems for Contraction Maps
3.2 The Markov-Kakutani Theorem
3.3 An Extension of the Hahn-Banach Theorem
3.4 Banach's Generalized Limits
3.5 Invariant Means on Semigroups
3.6 The Schauder-Tychonoff Theorem
3.7 The Work of Leray and Schauder
3.8 Fixed-point Theorems in Ordered Sets
Chapter 4 Topological Duals of Certain Spaces: Radon Measures
4.1 Plan of Chapter
4.2 The Banach Spaces lp(T) and c0(T)
4.3 The Dual of K(T): Radon Measures
4.4 Some Examples
4.5 Integration Theory Associated with a Positive Measure
4.6 The Space L1 = L1(T, µ) of µ-integrable Functions
4.7 Integrable Sets, Measurable Sets, Locally Negligible Sets
4.8 Measurable Functions
4.9 The Sup port of a Measure
4.10 The Space C(T) and Its Dual
4.11 The Seminorms Np
Inequalities of Hölder and Minkowski
the Spaces Lp and Lp
4.12 The Space L2 and Its Dual
4.13 Product of a Measure by a Function
4.14 Essentially Integrable Functions
4.15 The Lebesgue-Radon-Nikodym Theorem
4.16 The Dual of Lp and of Lp(1 = p 8)
4.17 Product Measures and the Fubini Theorem
4.18 Locally Compact Groups and Haar Measures
4.19 Group Algebras and Convolution
4.20 Weil's Criterion for Compactness in Lp Spaces over a Group
4.21 Weak Compactness in L1
the Dunford-Pettis Criterion
4.22 Weakly Compact Sets of Bounded Measures
4.23 A Theorem of Kakutani and Its Consequences
4.24 Remarks on Convexity Theorems
Chapter 5 Distributions and Linear Partial Differential Equations
5.0 Foreword
5.1 Distributions
5.2 Measures and Functions as Distributions
5.3 Convergence of Distributions
5.4 Differentiation of Distributions
5.5 Multiplication and Division of a Distribution by a Function. Pseudo functions
5.6 Restriction of Distributions
Localization
Support of a Distribution
5.7 Distributions with Compact Supports
5.8 Distributions of Finite Order
5.9 Distributions Whose First Derivatives Are Known
5.10 Convolution of Distributions on Rn
5.11 Further Properties of Distributions
5.12 Linear Partial Differential Equations
5.13 The Method of Orthogonal Projection and Its Developments
5.14 Existence of an Elementary Solution
Discussion of Problem 2
5.15 Fourier Transforms of Distributions
5.16 Discussion of Problem 1
5.17 Solution of Problem 3
5.18 Further Properties of Elementary Solutions
Hypoelliptic and Hyperbolic Operators and Equations
5.19 Characterization of Certain Types of Linear Partial Differential Operators by Inequalities and Order Relations
5.20 The Work of F. E. Browder
Chapter 6 The Open Mapping and Closed Graph Theorems
6.0 Foreword
6.1 Semimetrizable Topological Vector Spaces and Frechet Spaces
6.2 Barrelled, Infrabarrelled, and Ultrabarrelled Spaces
6.3 Inductive Limit Spaces
6.4 Forms of the Open Mapping and Closed Graph Theorems
6.5 Some Results Concerning Sequences of Complete Semimetrizable Topological Vector Spaces
6.6 Bounded Sets in Dm(O) and in K(T)
6.7 Another Version of the Closed Graph Theorem
6.8 Applications to Topological Bases
Chapter 7 Boundedness Principles
7.0 Foreword
7.1 Boundedness Principles for Barrelled and Ultrabarrelled Spaces
7.2 Some Applications and Examples
7.3 Bornological Spaces
7.4 Results for Sequentially Complete and Quasi-complete Spaces
7.5 Boundedness Principles for Complete Semimetrizable Spaces
7.6 An Application of Theorem 7.5.1 to Summation of Fourier Series
7.7 Boundedness Principles for Bilinear Maps
7.8 Some Applications
Chapter 8 Duality Theory
8.0 Foreword
8.1 Dual Systems and Weak Topologies
8.2 Properties of the Weakened Topology on a Locally Convex Space
8.3 Topologies Compatible with a Given Duality
8.4 The Strong Topology ß(E´, E) and Reflexivity. Montel Spaces
8.5 The Theorem of Banach-Grothendieck
8.6 Transposed and Adjoint Maps
8.7 The Bidual Space and the Biadjoint Mapping
8.8 Some Applications
8.9 Return to the Open Mapping and Closed Graph Theorems
8.10 Concerning Fully Complete Spaces
8.11 The Hellinger-Toeplitz Theorem
8.12 Concerning Weak Compactness
8.13 The Theorem of Krein
8.14 Integration of Vector-valued Functions
8.15 The Case in Which E is Metrizable
8.16 Functions with Values in E´. The (GDF) Property
8.17 The Dunford-Pettis Theorem
8.18 The Spaces LpE and LpE
8.19 Vector-valued Measures
8.20 The Dual of LpE when E Is a Banach Space and 1 & p 8
Chapter 9 Theory of Compact Operators
9.0 Foreword
9.1 Compact and Precompact Sets
9.2 Definition and First Properties of Compact and Weakly Compact Linear Maps
9.3 A Theorem about Weakly Compact Linear Maps
9.4 The Dunford-Pettis and Dieudonné Properties: Weakly Compact and Compact Linear Maps of Spaces L1 and C
9.5 Integral Operators and Kernel Representations
9.6 Return to the Theory of Compact Linear Maps
9.7 Endomorphism of Vector Spaces
9.8 Eigenvalues and Spectra
9.9 Results Concerning Spectra of Endomorphisms
9.10 Spectral Theory of Compact Endomorphisms
9.11 Spectral Theory of Compact Endomorphisms of a Hilbert Space
9.12 Partial Differential Equations and Compact Linear Maps
9.13 Return to Ergodic Theorems
Chapter 10 The Krein-Milman Theorem and Its Applications
10.0 Foreword
10.1 Extreme Points and the Krein-Milman Theorem
10.2 Application to Bernstein's Theorem
10.3 Application to Bochner's Theorem
10.4 The Plancherel Theorem and the Duality Law
Bibliography
Symbols and Abbreviations
Appendix: Additions to The Dover Edition
General Index

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