Series representations in spaces of vector‐valued functions via Schauder decompositions

It is a classical result that every C‐valued holomorphic function has a local power series representation. This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space E over C. Motivated by this example we try to answer the following question. Let E be a locally convex Hausdorff space over a field K, let F(Ω) be a locally convex Hausdorff space of K‐valued functions on a set Ω and let F(Ω,E) be an E‐valued counterpart of F(Ω) (where the term E‐valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of F(Ω) to elements of F(Ω,E)? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of functions F(Ω) having an equicontinuous Schauder basis.