Separability in function spaces (Q1082580)

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.