Topological Vector Spaces

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Springer Science & Business Media, Jul 1, 1999 - Mathematics - 346 pages
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The present book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra. The author has found it unnecessary to rederive these results, since they are equally basic for many other areas of mathematics, and every beginning graduate student is likely to have made their acquaintance. Simi larly, the elementary facts on Hilbert and Banach spaces are widely known and are not discussed in detail in this book, which is :plainly addressed to those readers who have attained and wish to get beyond the introductory level. The book has its origin in courses given by the author at Washington State University, the University of Michigan, and the University of Ttibingen in the years 1958-1963. At that time there existed no reasonably ccmplete text on topological vector spaces in English, and there seemed to be a genuine need for a book on this subject. This situation changed in 1963 with the appearance of the book by Kelley, Namioka et al. [1] which, through its many elegant proofs, has had some influence on the final draft of this manuscript. Yet the two books appear to be sufficiently different in spirit and subject matter to justify the publication of this manuscript; in particular, the present book includes a discussion of topological tensor products, nuclear spaces, ordered topological vector spaces, and an appendix on positive operators.
  

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Contents

PREREQUISITES
1
B GENERAL TOPOLOGY
4
C LINEAR ALGEBRA
9
TOPOLOGICAL VECTOR SPACES
12
2 PRODUCT SPACES SUBSPACES DIRECT SUMS QUOTIENT SPACES
19
3 TOPOLOGICAL VECTOR SPACES OF FINITE DIMENSION
21
4 LINEAR MANIFOLDS AND HYPERPLANES
24
5 BOUNDED SETS
25
4 DUALITY OF PROJECTIVE AND INDUCTIVE TOPOLOGIES
133
5 STRONG DUAL OF A LOCALLY CONVEX SPACE BIDUAL REFLEXIVE SPACES
140
6 DUAL CHARACTERIZATION OF COMPLETENESS METRIZABLE SPACES THEOREMS OF GROTHENDIECK BANACHDIEUDONNE AND...
147
7 ADJOINTS OF CLOSED LINEAR MAPPINGS
155
8 THE GENERAL OPEN MAPPING AND CLOSED GRAPH THEOREMS
161
9 TENSOR PRODUCTS AND NUCLEAR SPACES
167
10 NUCLEAR SPACES AND ABSOLUTE SUMMABILITY
176
11 WEAK COMPACTNESS THEOREMS OF EBERLEIN AND KREIN
185

6 METRIZABILITY
28
7 COMPLEXIFICATION
31
EXERCISES
33
LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
36
1 CONVEX SETS AND SEMINORMS
37
2 NORMED AND NORMABLE SPACES
40
3 THE HAHNBANACH THEOREM
45
4 LOCALLY CONVEX SPACES
47
5 PROJECTIVE TOPOLOGIES
51
6 INDUCTIVE TOPOLOGIES
54
7 BARRELED SPACES
60
8 BORNOLOGICAL SPACES
61
9 SEPARATION OF CONVEX SETS
63
10 COMPACT CONVEX SETS
66
EXERCISES
68
LINEAR MAPPINGS
73
1 CONTINUOUS LINEAR MAPS AND TOPOLOGICAL HOMOMORPHISMS
74
2 BANACHS HOMOMORPHISM THEOREM
76
3 SPACES OF LINEAR MAPPINGS
79
4 EQUICONTINUITY THE PRINCIPLE OF UNIFORM BOUNDEDNESS AND THE BANACHSTEINHAUS THEOREM
82
5 BILINEAR MAPPINGS
87
6 TOPOLOGICAL TENSOR PRODUCTS
92
7 NUCLEAR MAPPINGS AND SPACES
97
8 EXAMPLES OF NUCLEAR SPACES
106
9 THE APPROXIMATION PROPERTY COMPACT MAPS
108
EXERCISES
115
DUALITY
122
1 DUAL SYSTEMS AND WEAK TOPOLOGIES
123
2 ELEMENTARY PROPERTIES OF ADJOINT MAPS
128
3 LOCALLY CONVEX TOPOLOGIES CONSISTENT WITH A GIVEN DUALITY THE MACKEYARENS THEOREM
130
EXERCISES
190
ORDER STRUCTURES
203
1 ORDERED VECTOR SPACES OVER THE REAL FIELD
204
2 ORDERED VECTOR SPACES OVER THE COMPLEX FIELD
214
3 DUALITY OF CONVEX CONES
215
4 ORDERED TOPOLOGICAL VECTOR SPACES
222
5 POSITIVE LINEAR FORMS AND MAPPINGS
225
6 THE ORDER TOPOLOGY
230
7 TOPOLOGICAL VECTOR LATTICES
234
8 CONTINUOUS FUNCTIONS ON A COMPACT SPACE THEOREMS OF STONEWEIERSTRASS AND KAKUTANI
242
EXERCISES
250
C AND WALGEBRAS
258
1 PRELIMINARIES
259
2 CALGEBRAS THE GELFAND THEOREM
260
3 ORDER STRUCTURE OF A CALGEBRA
267
4 POSITIVE LINEAR FORMS REPRESENTATIONS
270
5 PROJECTIONS AND EXTREME POINTS
274
6 WALGEBRAS
277
7 VON NEUMANN ALGEBRAS KAPLANSKYS DENSITY THEOREM
287
8 PROJECTIONS AND TYPES OF WALGEBRAS
292
EXERCISES
299
SPECTRAL PROPERTIES OF POSITIVE OPERATORS
306
1 ELEMENTARY PROPERTIES OF THE RESOLVENT
307
2 PRINGSHEIMS THEOREM AND ITS CONSEQUENCES
309
3 THE PERIPHERAL POINT SPECTRUM
316
INDEX OF SYMBOLS
325
BIBLIOGRAPHY
330
INDEX
341
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About the author (1999)

Schaefer, Eberhard-Karls-Universtitat, Tubingen, Germany.

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