Lifting of holomorphic mappings on locally convex spaces (Q1092379)

If E and G are locally convex spaces, when can a holomorphic mapping f from E into a quotient space G/H be lifted to a holomorphic mapping \(\hat f: E\to G?\) A locally convex space F is said to have the (weak) holomorphic lifting property if for each locally convex space E and each quotient space \(G/H\simeq F\) for which each bounded subset in \(G/H\) can be lifted to a bounded subset in G, every holomorphic mapping \(f: E\to G/H\) of locally uniform bounded type has a lifting \(f: E\to G\) (resp. \(f: E\to G_ n''\)) of the same kind, where \(G_ n''\) denotes the bidual of G equipped with the natural topology. The author shows that each locally complete co-\({\mathcal F}_ 1\)-space (co-\({\mathcal L}_ 1\)-space) has the (weak) holomorphic lifting property. Under additional assumptions he shows that conversely a locally complete space with the (weak) holomorphic lifting property is a co-\({\mathcal F}_ 1\)-space (co-\({\mathcal L}_ 1\)-space). As a corollary he obtains the following. A (B)-space E is isomorphic to \(\ell_ 1(I)\) for some index I if and only if E has the holomorphic lifting property. A (B)-space is an \({\mathcal L}_ 1\)-space if and only if E has the weak holomorphic lifting property. If F is a (DFM)-space and G is a (FN)-space, then every holomorphic mapping f from E into every quotient space \(G/H\) has a holomorphic lifting \(\hat f: E\to G\). Applications to echelon and co- echelon spaces are given.