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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.

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