Locally Convex Spaces
(Sprache: Englisch)
The present book grew out of several courses which I have taught at the University of Zürich and at the University of Maryland during the past seven years. It is primarily intended to be a systematic text on locally convex spaces at the level of a student...
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The present book grew out of several courses which I have taught at the University of Zürich and at the University of Maryland during the past seven years. It is primarily intended to be a systematic text on locally convex spaces at the level of a student who has some familiarity with general topology and basic measure theory. However, since much of the material is of fairly recent origin and partly appears here for the first time in a book, and also since some well-known material has been given a not so well-known treatment, I hope that this book might prove useful even to more advanced readers. And in addition I hope that the selection ofmaterial marks a sufficient set-offfrom the treatments in e.g. N. Bourbaki [4], [5], R.E. Edwards [1], K. Floret-J. Wloka [1], H.G.Garnir-M.De Wilde-J.Schmets [1], AGrothendieck [13], H.Heuser [1], J. Horvath [1], J. L. Kelley-I. Namioka et al. [1], G. Köthe [7], [10], A P. Robertson W.Robertson [1], W.Rudin [2], H.H.Schaefer [1], F.Treves [l],A Wilansky [1]. A few sentences should be said about the organization of the book. It consists of 21 chapters which are grouped into three parts. Each chapter splits into several sections. Chapters, sections, and the statements therein are enumerated in consecutive fashion.
Inhaltsverzeichnis zu „Locally Convex Spaces “
I: Linear Topologies1 Vector Spaces
1.1 Generalities
1.2 Elementary Constructions
1.3 Linear Maps
1.4 Linear Independence
1.5 Linear Forms
1.6 Bilinear Maps and Tensor Products
1.7 Some Examples
2 Topological Vector Spaces
2.1 Generalities
2.2 Circled and Absorbent Sets
2.3 Bounded Sets. Continuous Linear Forms
2.4 Projective Topologies
2.5 A Universal Characterization of Products
2.6 Projective Limits
2.7 F-Seminorms
2.8 Metrizable Tvs
2.9 Projective Representation of Tvs
2.10 Linear Topologies on Function and Sequence Spaces
2.11 References
3 Completeness
3.1 Some General Concepts
3.2 Some Completeness Concepts
3.3 Completion of a Tvs
3.4 Extension of Uniformly Continuous Maps
3.5 Precompact Sets
3.6 Examples
3.7 References
4 Inductive Linear Topologies
4.1 Generalities
4.2 Quotients of Tvs
4.3 Direct Sums
4.4 Some Completeness Results
4.5 Inductive Limits
4.6 Strict Inductive Limits
4.7 References
5 Baire Tvs and Webbed Tvs
5.1 Baire Category
5.2 Webs in Tvs
5.3 Stability Properties of Webbed Tvs
5.4 The Closed Graph Theorem
5.5 Some Consequences
5.6 Strictly Webbed Tvs
5.7 Some Examples
5.8 References
6 Locally r-Convex Tvs
6.1 r-Convex Sets
6.2 r-Convex Sets in Tvs
6.3 Gauge Functionals and r-Seminorms
6.4 Continuity Properties of Gauge Functionals
6.5 Definition and Basic Properties of Lc,s
6.6 Some Permanence Properties of Lc,s
6.7 Bounded, Precompact, and Compact Sets
6.8 Locally Bounded Tvs
6.9 Linear Mappings Between r-Normable Tvs
6.10 Examples
6.11 References
7 Theorems of Hahn-Banach, Krein-Milman, and Riesz
7.1 Sublinear Functionals
7.2 Extension Theorem for Lcs
7.3 Separation Theorems
7.4 Extension Theorems for Normed Spaces
7.5 The Krein-Milman Theorem
7.6 The Riesz Representation Theorem
7.7 References
II: Duality Theory for Locally Convex Spaces
8 Basic Duality Theory
8.1 Dual Pairings and Weak Topologies
8.2 Polarization
8.3 Barrels and Disks
8.4 Bornologies and
... mehr
?-Topologies
8.5 Equicontinuous Sets and Compactologies
8.6 Continuity of Linear Maps
8.7 Duality of Subspaces and Quotients
8.8 Duality of Products and Direct Sums
8.9 The Stone-Weierstrass Theorem
8.10 References
9 Continuous Convergence and Related Topologies
9.1 Continuous Convergence
9.2 Grothendieck's Completeness Theorem
9.3 The Topologies ?t and ?
9.4 The Banach-Dieudonné Theorem
9.5 B-Completeness and Related Properties
9.6 Open and Nearly Open Mappings
9.7 Application to B-Completeness
9.8 On Weak Compactness
9.9 References
10 Local Convergence and Schwartz Spaces
10.1 ?-Convergence. Local Convergence
10.2 Local Completeness
10.3 Equicontinuous Convergence. The Topologies ?t and ?
10.4 Schwartz Topologies
10.5 A Universal Schwartz Space
10.6 Diametral Dimension. Power Series Spaces
10.7 Quasi-Normable Lcs
10.8 Application to Continuous Function Spaces
10.9 References
11 Barrelledness and Reflexivity
11.1 Barrelled Lcs
11.2 Quasi-Barrelled Lcs
11.3 Some Permanence Properties
11.4 Semi-Reflexive and Reflexive Lcs
11.5 Semi-Montel and Montei Spaces
11.6 On Fréchet-Montel Spaces
11.7 Application to Continuous Function Spaces
11.8 On Uniformly Convex Banach Spaces
11.9 On Hilbert Spaces
11.10 References
12 Sequential Barrelledness
12.1 ??-Barrelled and c0-Barrelled Lcs
12.2 ?0-Barrelled Lcs
12.3 Absorbent and Bornivorous Sequences
12.4 DF-Spaces, gDF-Spaces, and df-Spaces
12.5 Relations to Schwartz Topologies
12.6 Application to Continuous Function Spaces
12.7 References
13 Bornological and Ultrabornological Spaces
13.1 Generalities
13.2 ?-Convergent and Rapidly ?-Convergent Sequences
13.3 Associated Bornological and Ultrabornological Spaces
13.4 On the Topology ?(E', E)bor
13.5 Permanence Properties
13.6 Application to Continuous Function Spaces
13.7 References
14 On Topological Bases
14.1 Biorthogonal Sequences
14.2 Bases and Schauder Bases
14.3 Weak Bases. Equicontinuous Bases
14.4 Examples and Additional Remarks
14.5 Shrinking and Boundedly Complete Bases
14.6 On Summable Sequences
14.7 Unconditional and Absolute Bases
14.8 Orthonormal Bases in Hilbert Spaces
14.9 References
III Tensor Products and Nuclearity
15 The Projective Tensor Product
15.1 Generalities on Projective Tensor Products
15.2 Tensor Product and Linear Mappings
15.3 Linear Mappings with Values in a Dual
15.4 Projective Limits and Projective Tensor Products
15.5 Inductive Limits and Projective Tensor Products
15.6 Some Stability Properties
15.7 Projective Tensor Products with ?1 (?)-spaces
15.8 References
16 The Injective Tensor Product
16.1 ?-Products and ?-Tensor Products
16.2 Tensor Product and Linear Mappings
16.3 Projective and Inductive Limits
16.4 Some Stability Properties
16.5 Spaces of Summable Sequences
16.6 Continuous Vector Valued Functions
16.7 Holomorphic Vector Valued Functions
16.8 References
17 Some Classes of Operators
17.1 Compact Operators
17.2 Weakly Compact Operators
17.3 Nuclear Operators
17.4 Integral Operators
17.5 The Trace for Finite Operators
17.6 Some Particular Cases
17.7 References
18 The Approximation Property
18.1 Generalities
18.2 Some Stability Properties
18.3 The Approximation Property for Banach Spaces
18.4 The Metric Approximation Property
18.5 The Approximation Property for Concrete Spaces
18.6 References
19 Ideals of Operators in Banach Spaces
19.1 Generalities
19.2 Dual, Injective, and Surjective Ideals
19.3 Ideal-Quasinorms
19.4 ?p-Sequences
19.5 Absolutely p-Summing Operators
19.6 Factorization
19.7 p-Nuclear Operators
19.8 p-Approximable Operators
19.9 Strongly Nuclear Operators
19.10 Some Multiplication Theorems
19.11 References
20 Components of Ideals on Particular Spaces
20.1 Compact Operators on Hilbert Spaces
20.2 The Schatten-von Neumann Classes
20.3 Grothendieck's Inequality
20.4 Applications
20.5$$ {P_p}and{N_q} $$on Hilbert Spaces
20.6 Composition of Absolutely Summing Operators
20.7 Weakly Compact Operators on T(K)-Spaces
20.8 References
21 Nuclear Locally Convex Spaces
21.1 Locally Convex A-Spaces
21.2 Generalities on Nuclear Spaces
21.3 Further Characterizations by Tensor Products
21.4 Nuclear Spaces and Choquet Simplexes
21.5 On Co-Nuclear Spaces
21.6 Examples of Nuclear Spaces
21.7 A Universal Generator
21.8 Strongly Nuclear Spaces
21.9 Associated Topologies
21.10 Bases in Nuclear Spaces
21.11 References
- List of Symbols
8.5 Equicontinuous Sets and Compactologies
8.6 Continuity of Linear Maps
8.7 Duality of Subspaces and Quotients
8.8 Duality of Products and Direct Sums
8.9 The Stone-Weierstrass Theorem
8.10 References
9 Continuous Convergence and Related Topologies
9.1 Continuous Convergence
9.2 Grothendieck's Completeness Theorem
9.3 The Topologies ?t and ?
9.4 The Banach-Dieudonné Theorem
9.5 B-Completeness and Related Properties
9.6 Open and Nearly Open Mappings
9.7 Application to B-Completeness
9.8 On Weak Compactness
9.9 References
10 Local Convergence and Schwartz Spaces
10.1 ?-Convergence. Local Convergence
10.2 Local Completeness
10.3 Equicontinuous Convergence. The Topologies ?t and ?
10.4 Schwartz Topologies
10.5 A Universal Schwartz Space
10.6 Diametral Dimension. Power Series Spaces
10.7 Quasi-Normable Lcs
10.8 Application to Continuous Function Spaces
10.9 References
11 Barrelledness and Reflexivity
11.1 Barrelled Lcs
11.2 Quasi-Barrelled Lcs
11.3 Some Permanence Properties
11.4 Semi-Reflexive and Reflexive Lcs
11.5 Semi-Montel and Montei Spaces
11.6 On Fréchet-Montel Spaces
11.7 Application to Continuous Function Spaces
11.8 On Uniformly Convex Banach Spaces
11.9 On Hilbert Spaces
11.10 References
12 Sequential Barrelledness
12.1 ??-Barrelled and c0-Barrelled Lcs
12.2 ?0-Barrelled Lcs
12.3 Absorbent and Bornivorous Sequences
12.4 DF-Spaces, gDF-Spaces, and df-Spaces
12.5 Relations to Schwartz Topologies
12.6 Application to Continuous Function Spaces
12.7 References
13 Bornological and Ultrabornological Spaces
13.1 Generalities
13.2 ?-Convergent and Rapidly ?-Convergent Sequences
13.3 Associated Bornological and Ultrabornological Spaces
13.4 On the Topology ?(E', E)bor
13.5 Permanence Properties
13.6 Application to Continuous Function Spaces
13.7 References
14 On Topological Bases
14.1 Biorthogonal Sequences
14.2 Bases and Schauder Bases
14.3 Weak Bases. Equicontinuous Bases
14.4 Examples and Additional Remarks
14.5 Shrinking and Boundedly Complete Bases
14.6 On Summable Sequences
14.7 Unconditional and Absolute Bases
14.8 Orthonormal Bases in Hilbert Spaces
14.9 References
III Tensor Products and Nuclearity
15 The Projective Tensor Product
15.1 Generalities on Projective Tensor Products
15.2 Tensor Product and Linear Mappings
15.3 Linear Mappings with Values in a Dual
15.4 Projective Limits and Projective Tensor Products
15.5 Inductive Limits and Projective Tensor Products
15.6 Some Stability Properties
15.7 Projective Tensor Products with ?1 (?)-spaces
15.8 References
16 The Injective Tensor Product
16.1 ?-Products and ?-Tensor Products
16.2 Tensor Product and Linear Mappings
16.3 Projective and Inductive Limits
16.4 Some Stability Properties
16.5 Spaces of Summable Sequences
16.6 Continuous Vector Valued Functions
16.7 Holomorphic Vector Valued Functions
16.8 References
17 Some Classes of Operators
17.1 Compact Operators
17.2 Weakly Compact Operators
17.3 Nuclear Operators
17.4 Integral Operators
17.5 The Trace for Finite Operators
17.6 Some Particular Cases
17.7 References
18 The Approximation Property
18.1 Generalities
18.2 Some Stability Properties
18.3 The Approximation Property for Banach Spaces
18.4 The Metric Approximation Property
18.5 The Approximation Property for Concrete Spaces
18.6 References
19 Ideals of Operators in Banach Spaces
19.1 Generalities
19.2 Dual, Injective, and Surjective Ideals
19.3 Ideal-Quasinorms
19.4 ?p-Sequences
19.5 Absolutely p-Summing Operators
19.6 Factorization
19.7 p-Nuclear Operators
19.8 p-Approximable Operators
19.9 Strongly Nuclear Operators
19.10 Some Multiplication Theorems
19.11 References
20 Components of Ideals on Particular Spaces
20.1 Compact Operators on Hilbert Spaces
20.2 The Schatten-von Neumann Classes
20.3 Grothendieck's Inequality
20.4 Applications
20.5$$ {P_p}and{N_q} $$on Hilbert Spaces
20.6 Composition of Absolutely Summing Operators
20.7 Weakly Compact Operators on T(K)-Spaces
20.8 References
21 Nuclear Locally Convex Spaces
21.1 Locally Convex A-Spaces
21.2 Generalities on Nuclear Spaces
21.3 Further Characterizations by Tensor Products
21.4 Nuclear Spaces and Choquet Simplexes
21.5 On Co-Nuclear Spaces
21.6 Examples of Nuclear Spaces
21.7 A Universal Generator
21.8 Strongly Nuclear Spaces
21.9 Associated Topologies
21.10 Bases in Nuclear Spaces
21.11 References
- List of Symbols
... weniger
Bibliographische Angaben
- 1981, 552 Seiten, Maße: 17,8 x 25,4 cm, Kartoniert (TB), Englisch
- Verlag: Vieweg & Teubner
- ISBN-10: 3322905616
- ISBN-13: 9783322905611
- Erscheinungsdatum: 11.04.2014
Sprache:
Englisch
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