AbstractIt is a classical result that every ‐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 spaceEover . Motivated by this example we try to answer the following question. LetEbe a locally convex Hausdorff space over a field , let be a locally convex Hausdorff space of ‐valued functions on a set Ω and let be anE‐valued counterpart of (where the termE‐valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of to elements of ? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of functions having an equicontinuous Schauder basis.