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We construct a countable inductive limit of weighted Banach spaces ofholomorphic functions, which is not a topological subspace of thecorresponding weighted inductive limit of spaces of continuous functions. Themain step of our construction, using a special sequence of outer holomorphicfunctions, shows that a certain sequence space is isomorphic to a complementedsubspace of a weighted space of holomorphic functions in two complexvariables.  This example solves in the negative a well-known open problem raised byBierstedt, Meise and Summers.

The subspace problem for weighted inductive limits of spaces of holomorphic functions.