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 Hw(U ; F ) of holomorphic functions defined on an open subset U of a Banach space E with values in a Banach space F. After having introduced a locally convex topology τM on the space Hw(U;F), we show that (Hw(U;F), τM) is topologically isomorphic to (L(Gw(U);F), τc) where Gw(U) is the predual of Hw(U) consisting of all linear functionals whose restrictions to the closed unit ball of Hw(U) are continuous for the compact open topology τ0 . Finally, these results have been used in characterizing the approximation property for the space Hw(U) and its predual for a suitably restricted weight w.

Keywords

• Holomorphic mappings
• weighted spaces of holomorphic functions
• linearization
• approximation property