Uniqueness of the Topology on Spaces of Vector-Valued Functions

Let be a topological space without isolated points, let E bea topological linear space which is continuously embedded intoa product of countably boundedly generated topological linearspaces, and let X be a linear subspace of C(, E). If a C()is not constant on any open subset of and aX X, then it isshown that there is at most one F-space topology on X that makesthe multiplication by a continuous. Furthermore, if U is a subsetof C() which separates strongly the points of and UX X, thenit is proved that there is at most one F-space topology on Xthat makes the multiplication by a continuous for each a U. These results are applied to the study of the uniqueness ofthe F-space topology and the continuity of translation invariantoperators on the Banach space L₁(G, E) for a noncompact locallycompact group G and a Banach space E. Furthermore, the problemsof the uniqueness of the F-algebra topology and the continuityof epimorphisms and derivations on F-algebras and some algebrasof vector-valued functions are considered.