Weighted holomorphic mappings attaining their norms

Given an open subset U of C-n, a weight v on U and a complex Banach space F, let H-v (U, F) denote the Banach space of all weighted holomorphic mappings f : U -> F, under the weighted supremum norm parallel to f parallel to(v) := sup {v (z) parallel to f(z)parallel to : z is an element of U}. We prove that the set of all mappings f is an element of H-v (U, F) that attain their weighted supremum norms is norm dense in H-v (U, F), provided that the closed unit ball of the little weighted holomorphic space H-v0(U, F) is compact-open dense in the closed unit ball of H-v (U, F). We also prove a similar result for mappings f is an element of H-v (U, F) such that vf has a relatively compact range.