Linearization of weighted (LB)-spaces of entire functions on Banach spaces

Abstract

Given a decreasing sequence of weights V on a Banach space X, we consider the weighted inductive limits of spaces of entire functions V H(X) and V H 0(X). We prove that V H(X) is the strong dual of a Frechet space F for a particular class of sequence of weights, and we study some conditions to ensure that the equality V H 0(X)′′ = V H(X) holds. The existence of a predual of V H(X) leads to a linearization of this space of holomorphic functions. In fact, we show that there exists a holomorphic function $${\Delta: X \rightarrow F}$$ with the following universal property: for each Banach space E and each function $${f \in V H(X, E),}$$ there is a unique continuous linear operator $${T_f: F\rightarrow E}$$ such that $${T_f \circ \Delta = f.}$$ The correspondence $${f\rightarrow T_f}$$ is an isomorphism between the space V H(X, E) and the space $${\mathcal{L}_i(F, E)}$$ of all continuous linear operators from F into E endowed with the inductive limit topology.