Normal families of holomorphic functions and mappings on a Banach space
A family \(\mathcal{F}\subset \mathcal{H}(\Omega)\) of holomorphic functions \(\Omega\rightarrow \mathbb C\) is called normal if every sequence \((f_n)\subset \mathcal{F}\) contains either (i) a subsequence converging uniformly on all compact sets \(K\subset \Omega\), or (ii) a subsequence `compactly diverging': given \(K\subset \Omega\) and \(K' \subset \mathbb C\), \(f_{n_j}(K)\cap K'=\emptyset\) for all large enough \(j\). The following version of Montel's theorem is well-known in infinite-dimensinal holomorphy (see Lemma 3.25 in [\textit{S. Dineen}, ``Complex analysis on infinite-dimensional spaces'' (Springer Monographs in Mathematics, Springer-Verlag, London) (1999; Zbl 1034.46504)]): If \(\mathcal{F}\subset \mathcal{H}(\Omega)\) is locally bounded, then \(\mathcal{F}\) is relatively compact in the topology of uniform convergence on compact sets. In the paper under review, the authors discuss other Montel-type theorems, including: If \(\mathcal{F}\subset \mathcal{H}(\Omega)\) is uniformly bounded on compact sets, then given any sequence in \(\mathcal{F}\) and given \(K\subset \Omega\) compact, there is a subsequence converging uniformly on \(K\). Perhaps more importantly, the notion of normality in infinite dimensions is extensively discussed and presented from several different viewpoints, including: comparison of different topologies on \(\mathcal{H}(\Omega)\), currents, invariant metrics, normal functions, and functional analysis. The paper is very nicely written and includes several illuminating examples.
scientific article; zbMATH DE number 2065404
scientific article