## Topological Vector SpacesThe 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 |

325 | |

330 | |

341 | |

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### Common terms and phrases

0-neighborhood base adjoint algebra assertion Banach lattice Banach space bilinear bounded sets bounded subsets C*-algebra called canonical map canonical order Chapter circled 0-neighborhood circled hull circled subset closed subspace closure compact space compact subset contains continuous linear form continuous linear map convex hull Corollary countable defined denote dense direct sum duality elements equicontinuous equicontinuous subsets equivalent example Exercise exists F)-space filter finite subset follows functions Hausdorff t.v.s. hence Hilbert space hyperplane I.c. topology implies inductive limit isomorphism lemma let F locally convex space locally convex topology metrizable non-empty normal cone normed space nuclear spaces order complete order interval order topology ordered vector space positive cone positive linear form precompact projective topology properties quasi-complete reflexive respect S-topology satisfying Section semi-norms semi-reflexive separated spectral strong dual subspace summable theorem unique unit ball vector lattice weakly