Weighted spaces of holomorphic functions on Banach spaces and the approximation property

Abstract

In this paper, we study the linearization theorem for the weighted space H_ω(U; F) of holomorphic functions de_ned on an open subset U of a Banach space E with values in a Banach space F. After having introduced a locally convex topology T_M on the space H_w (U; F), we show that (H_w (U; F);T_M) is topologically isomorphic to (L(G_ω (U); F), T_c ) where G_w (U)is the predual of H_w(U) consisting of all linear functionals whose restrictions to the closed unit ball of H_w(U) are continuous for the compact open topology T_0. Finally, these results have been used in characterizing the approximation property for the space H_w(U) and its predual for a suitably restricted weight w.