Topological Vector Spaces ICUP Archive, 1983 - 456 pages It is the author's aim to give a systematic account of the most im portant ideas, methods and results of the theory of topological vector spaces. After a rapid development during the last 15 years, this theory has now achieved a form which makes such an account seem both possible and desirable. This present first volume begins with the fundamental ideas of general topology. These are of crucial importance for the theory that follows, and so it seems necessary to give a concise account, giving complete proofs. This also has the advantage that the only preliminary knowledge required for reading this book is of classical analysis and set theory. In the second chapter, infinite dimensional linear algebra is considered in comparative detail. As a result, the concept of dual pair and linear topologies on vector spaces over arbitrary fields are intro duced in a natural way. It appears to the author to be of interest to follow the theory of these linearly topologised spaces quite far, since this theory can be developed in a way which closely resembles the theory of locally convex spaces. It should however be stressed that this part of chapter two is not needed for the comprehension of the later chapters. Chapter three is concerned with real and complex topological vector spaces. The classical results of Banach's theory are given here, as are fundamental results about convex sets in infinite dimensional spaces. |
Table des matières
Preface page | 1 |
Duality and the HahnBanach theorem | 23 |
Topologies on dual spaces and the MackeyArens | 44 |
Barrelled spaces and the BanachSteinhaus | 65 |
Inductive and projective limits | 76 |
Completeness and the closed graph theorem | 101 |
Some further topics | 127 |
Compact linear mappings | 142 |
| 155 | |
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Expressions et termes fréquents
absolutely convex envelope absolutely convex neighbourhood absorbent algebraic Banach space barrelled barrelled space base of neighbourhoods belongs bilinear bounded sets bounded subsets bourhood called Cauchy Chapter clearly closed absolutely convex closed vector subspace closure compact set contained continuous linear form continuous linear mapping convergence convex topology COROLLARY defined denoted direct sum dual pair elements equicontinuous equivalent example fact filter finer finite follows Fréchet space fully complete functions give given graph Hence identity implies intersection isomorphism Lemma meeting metrisable nearly normed space origin polar precompact projective limit Proof Prop PROPOSITION prove quotient result satisfied scalar seminorm separated convex space sequence space E space F spans Suppl Suppose taking theorem theory topological space topology of uniform transpose uniform convergence valued vector space vector subspace