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 [Ufr ] is a subset of C(Ω) which separates strongly the points
of Ω and [Ufr ]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 ∈ [Ufr ].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 L1(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.
Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces
