Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

We prove the following result which extends in a somewhat “linear” sense a theorem by Kierst and Szpilrajn and which holds on many “natural” spaces of holomorphic functions in the open unit disk ${\mathbb D}$: There exist a dense linear manifold and a closed infinite-dimensional linear manifold of holomorphic functions in ${\mathbb D}$ whose domain of holomorphy is ${\mathbb D}$ except for the null function. The existence of a dense linear manifold of noncontinuable functions is also shown in any domain for its full space of holomorphic functions.