Entire functions on Banach spaces with a separable dual (Q1614763)

Let \(E\) and \(F\) be complex Banach spaces and let \(H(E;F)\) be the space of all holomorphic functions from \(E\) into \(F\). Let \(H_{wu}(E;F)\) (resp. \(H_{w}(E;F)\), resp. \(H_{wsc}(E;F)\), resp. \(H_{bk}(E;F)\), resp. \(H_{b}(E;F)\)) be the subspace of all \(f \in H(E;F)\) which are weakly uniformly continuous on bounded sets (resp. weakly continuous on bounded sets, resp. weakly sequentially continuous, resp. bounded on weakly compact sets, resp. bounded on bounded sets). In \textit{R. M. Aron, C. Herves}, and \textit{M. Valdivia} [J. Funct. Anal. 17, 189-203 (1983; Zbl 0517.46019)] the following questions appeared. (A) Does \(H_{w}(E;F)= H_{wu}(E;F)\) for arbitrary \(E\) and \(F\)? (B) Does \(H_{wsc}(E;F)=H_{wu}(E;F)\) when \(E\) has a separable dual and \(F\) arbitrary? In this paper the authors give an affirmative answer to (B). Moreover, they extend a result of \textit{S. Dineen} [J. Funct. Anal. 52, 205-218 (1983; Zbl 0538.46032)] to show that (*) \(H_{bk}(E;F)= H_{b}(E;F)\) if \(E\) has a separable dual, thus showing equation (*) for spaces \(E\) with shrinking bases and using a quotient spaces argument. This implies that (B) gives a partial answer to (A). The paper includes all proofs.