The Subspace Problem for Weighted Inductive Limits of Spaces of Holomorphic Functions

The aim of the present article is to solve in the negative a well-known open problem raised by Bierstedt, Meise and Summers in [BMS1] (see also [BM1]). We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions. As a consequence the topology of the weighted inductive limit of spaces of holomorphic functions cannot be described by the weighted sup-seminorms given by the maximal system of weights associated with the sequence of weights defining the inductive limit. The main step of our construction shows that a certain sequence space is isomorphic to a complemented subspace of a weighted space of holomorphic functions. To do this we make use of a special sequence of outer holomorphic functions and of the existence of radial limits of holomorphic bounded functions in the disc.