Weighted holomorphic mappings attaining their norms

Given an open subset U of \({\mathbb {C}}^n,\) a weight v on U and a complex Banach space F,  let \(\mathcal {H}_v(U,F)\) denote the Banach space of all weighted holomorphic mappings \(f:U\rightarrow F,\) under the weighted supremum norm \(\left\| f\right\| _v:=\sup \left\{ v(z)\left\| f(z)\right\| :z\in U\right\} .\) We prove that the set of all mappings \(f\in \mathcal {H}_v(U,F)\) that attain their weighted supremum norms is norm dense in \(\mathcal {H}_v(U,F),\) provided that the closed unit ball of the little weighted holomorphic space \(\mathcal {H}_{v_0}(U,F)\) is compact-open dense in the closed unit ball of \(\mathcal {H}_v(U,F).\) We also prove a similar result for mappings \(f\in \mathcal {H}_v(U,F)\) such that vf has a relatively compact range.