Realcompactness and spaces of vector-valued functions
Copyright policy )Realcompactness and spaces of vector-valued functions
It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.
15 pages, LaTeX. Results stated for arbitrary normed spaces without changes in proofs. New presentation and new examples. One reference added
- University of Cantabria Spain
biseparating map, spaces of vector-valued functions, General Topology (math.GN), Algebraic properties of function spaces in general topology, homeomorphic compactifications, Functional Analysis (math.FA), Mathematics - Functional Analysis, Function spaces in general topology, Spaces of vector- and operator-valued functions, Realcompactness and realcompactification, FOS: Mathematics, 54C35 (Primary) 54C40, 54D60, 46E40 (Secondary), realcompactness, Mathematics - General Topology
biseparating map, spaces of vector-valued functions, General Topology (math.GN), Algebraic properties of function spaces in general topology, homeomorphic compactifications, Functional Analysis (math.FA), Mathematics - Functional Analysis, Function spaces in general topology, Spaces of vector- and operator-valued functions, Realcompactness and realcompactification, FOS: Mathematics, 54C35 (Primary) 54C40, 54D60, 46E40 (Secondary), realcompactness, Mathematics - General Topology
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