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Embedding in topological vector spaces

Gary Richardson

1977
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Proceedings of the American Mathematical Society
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Let LX denote the set of all continuous linear functional on the locally convex topological vector space X. The space L^X denotes LX endowed with the compact-open topology. We investigate the spaces, X, which have the property that the natural map from X into L^tL^X) is an embedding. Preliminaries. The reader is referred to Jarchow [6] for basic definitions and terminology not mentioned here. Since convergence spaces are more abundant than topological spaces, it is sometimes convenient to
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... convenient to characterize certain topological properties in terms of an associated convergence space. For example, the completion of a Hausdorff locally convex topological vector space X is shown by Butzmann [1] to be LC(LCX), where LCX denotes the set LX endowed with the continuous convergence structure; that is, the finest convergence structure, c, such that the evaluation map w: LCX X X -» 7? is continuous. Convergence space properties of LCX are used here to investigate the spaces, X, such that the natural map i: X -> LJ^L^X) is an embedding, where "co" denotes the compact-open topology. A space X satisfying the latter condition is called co-embedded. The symbol X will always denote a Hausdorff locally convex topological vector space. Another convenient convergence vector space associated with X is A", defined as the set X equipped with the following convergence structure: §" -»• x in A" iff ?F -» x in X and <$ contains an A"-compact subset. The space X" is the coarsest locally compact convergence vector space finer than X. Moreover, for any convergence vector space Y, let KY denote the finest locally convex topological vector space coarser than Y. Let kX denote the coarsest fc-space finer than X, and let ckX be the space obtained from kX by taking the convex neighborhoods of 0 in kX as a base for the neighborhood filter of 0 in ckX. These notions were introduced and studied by Frölicher and Jarchow [3] . A space X is called a ck-space whenever ckX = X. Moreover, ckX is the finest locally convex topological vector space which has the same compact subsets as X; furthermore, KX' = ckX. Hence ti follows that X is a c&-space iff X = KY for some locally compact convergence vector space

doi:10.1090/s0002-9939-1977-0442630-4
fatcat:3ieejoydabflvi3lfb5bpjiog4