Separability in function spaces
Copyright policy )Separability in function spaces
Let X be a completely regular Hausdorff space and \(E\) a real Hausdorff topological vector space, and let \(\beta_ 0\) and \(\kappa\) be the substrict and compact-open topologies on \(C_ b(X,E)\) and \(C(X,E)\), respectively. If \(E\) is separable with nontrivial dual, then the following are shown to be equivalent: (1) X is separably submetrizable; (2) \((C_ b(X)\otimes E,\beta_ 0)\) is separable; (3) \((C(X)\otimes E,\kappa)\) is separable. For E a locally bounded space, \((C_ b(X,E),\beta_ 0)\) is separable iff \((C(X,E),\kappa)\) is so.
- University of Benghazi Libyan Arab Jamahiriya
completely regular Hausdorff space, separably submetrizable, Applied Mathematics, compact-open topologies, vector-valued continuous functions, Spaces of vector- and operator-valued functions, Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.), Topological linear spaces of continuous, differentiable or analytic functions, shrinkable neighbourhood, locally bounded space, substrict topology, Analysis
completely regular Hausdorff space, separably submetrizable, Applied Mathematics, compact-open topologies, vector-valued continuous functions, Spaces of vector- and operator-valued functions, Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.), Topological linear spaces of continuous, differentiable or analytic functions, shrinkable neighbourhood, locally bounded space, substrict topology, Analysis
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