## Abstract

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

Original language | English |
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Pages (from-to) | 354-376 |

Number of pages | 23 |

Journal | Mathematische Nachrichten |

Volume | 294 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2021 |

Externally published | Yes |

## Keywords

- injective tensor product
- Schauder basis
- Schauder decomposition
- series representation
- vector-valued function