Weakly uniformly continuous holomorphic functions and the approximation property (Q1602597)

\textit{M.\,Schottenloher} and the reviewer [J.~Funct.\ Anal.\ 21, 7--30 (1976; Zbl 0328.46046)] showed the equivalence of the following properties for a complex Banach space \(E:\) \(E\) has the approximation property; for every nonempty open subset \(U \subset E,\) the space of holomorphic functions on \(U\), \({\mathcal H}(U)\), has the approximation property when endowed with the compact open topology; \({\mathcal H}(U) \otimes F\) is dense in the space \({\mathcal H}(U;F)\) of \(F\)-valued holomorphic functions on \(U \subset E\), for every \(U\) and every complex Banach space \(F\). Here, the authors apply \(\varepsilon\)-product techniques introduced by \textit{L.\,Schwartz} [J.~Anal.\ Math.\ 4, 88--148 (1955; Zbl 0066.09601)] to study analogous problems, primarily in the context of vector-valued weakly uniformly continuous holomorphic mappings from a complex locally convex space \(E\) to another such space \(F\). To do this, they first define \({\mathcal{H}}_{wu,a}(U;F)\) to be the space of Gâteaux holomorphic mappings that are weakly uniformly continuous on \(U\)-bounded subsets, and \({\mathcal H}_{wu}(U;F)\) to be the elements of \({\mathcal H}_{wu,a}(U;F)\) that are also holomorphic. One of their principal results is that if \(F\) is quasi-complete, then \({\mathcal H}_{wu,a}(U;F),\) endowed with the topology of uniform convergence on \(U\)-bounded sets, is topologically isomorphic to \({\mathcal H}_{wu,a}(U;F)\). Moreover, under reasonable conditions on \(E\) and \(F,\) they prove that the following properties are equivalent: \(E_b'\) has the approximation property; \({\mathcal H}_{wu}(U)\) has the approximation property for every balanced open \(U \subset E\); \({\mathcal H}_{wu}(U;F)\) has the approximation property for every balanced open \(U\) and every quasi-complete \(F\) having the approximation property.