Duality in spaces of Lorch analytic mappings (Q2833458)

A function \(f\) from a subset \(U\) of a Banach algebra \(E\) into \(E\) is said to be Lorch analytic if, for each \(a\) in \(U\), there is \(\rho>0\) and a sequence \((a_n)_n\) in \(E\) such that \(f(z)=\sum_{n=0}^\infty a_n(z-a)^n\) for all \(\|z-a\| <\rho\). The space of all Lorch analytic functions from \(U\) into \(E\) is denoted by \({\mathcal H}_L(E)\). When endowed with the topology of uniform convergence on bounded sets, \({\mathcal H}_L(E)\) becomes a Fréchet space. In this paper, the authors show that the strong dual of \({\mathcal H}_L(E)\) can be identified with a countable inductive limit of Banach sequence spaces. The strong and inductive duals of \({\mathcal H}_L (E)\) are shown to be equal. For each natural number \(n\), the authors use \({\mathcal A}_L(nB_E)\) to denote the Banach algebra of Lorch analytic functions on \(nB_E\) and show that the strong dual of \({\mathcal H}_L(E)\) can be identified with the inductive limit of duals of \({\mathcal A}(nB_E)'\).