A Canonical Extension for Analytic Functions on Banach Spaces

Abstract

Given Banach spaces $E$ and $F$, a Banach space ${G_{EF}}$ is presented in which $E$ is embedded and which seems a natural space to which extend $F$-valued analytic functions. Any $F$-valued analytic function defined on a subset $U$ of $E$ may be extended to an open neighborhood of $U$ in ${G_{EF}}$. This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.