Weighted (LB)-spaces of holomorphic functions: VH(G)=V0H(G) and completeness of V0H(G)

In this article we show that algebraic equalities between weighted inductive limits of spaces of holomorphic functions and/or between their projective hulls sometimes have strong consequences for the locally convex properties of the spaces involved in these equalities. For these results we impose the typical conditions which imply biduality between the spaces with the o- and O-growth conditions and use interpolating sequences for the step spaces Hvn(G), n∈N. In Section 1 we show that VH(G)=V0H(G) holds algebraically if and only if this space is (DFS) and that is true if and only if the space is semi-Montel. In Section 2 we provide a new characterization of the semi-Montel property of , which is much simpler than the one given before (in [K.D. Bierstedt, J. Bonet, J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998) 137–168]). Section 3 proves that the completeness of V0H(G) sometimes implies that the inductive limit indnH(vn)0(G) must be boundedly retractive.