Uniqueness of the Topology on Spaces of Vector‐Valued Functions

Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C (Ω, E ). If a ∈ C (Ω) is not constant on any open subset of Ω and aX ⊂ X , then it is shown that there is at most one F ‐space topology on X that makes the multiplication by a continuous. Furthermore, if U is a subset of C (Ω) which separates strongly the points of Ω and U X ⊂ X , then it is proved that there is at most one F ‐space topology on X that makes the multiplication by a continuous for each a ∈ U. These results are applied to the study of the uniqueness of the F ‐space topology and the continuity of translation invariant operators on the Banach space L 1 ( G , E ) for a noncompact locally compact group G and a Banach space E . Furthermore, the problems of the uniqueness of the F ‐algebra topology and the continuity of epimorphisms and derivations on F ‐algebras and some algebras of vector‐valued functions are considered.