Normal families of holomorphic functions on infinite-dimensional spaces.

*(English)*Zbl 1124.46025The article is based on the author’s doctoral dissertation with J. Mujica. Its aim is to extend some classical results of Montel, Hurwitz and Schottky from holomorphic functions in one (or a finite number of) variable(s) to the infinite-dimensional case.

Let \(U\) denote a nonvoid open subset of a complex locally convex space \(E\), and let \({\mathcal H}(U)\) be the space of all complex-valued holomorphic functions on \(U\), endowed with the topology of uniform convergence on the compact subsets of \(U\). In Section 2, it is proved that every locally bounded family \({\mathcal F} \subset {\mathcal H}(U)\) is normal if \(E\) is separable. The separability hypothesis is necessary here. Some related results are also given. Section 3 is devoted to the study of normal families of holomorphic functions with exceptional values. It is proved that, for a separable locally convex space \(E\) and for a connected open subset \(U\) of \(E\), a family \({\mathcal F}\) of holomorphic functions on \(U\) with two different exceptional values \(a\) and \(b\) (i.e., the functions in \({\mathcal F}\) omit the values \(a\) and \(b\)) is \(\mathbb{C}_\infty\)-normal (i.e., each sequence in \({\mathcal F}\) either admits a subsequence converging in \({\mathcal H}(U)\) with respect to the compact-open topology or admits a subsequence which diverges to infinity uniformly on each compact subset of \(U\)). An extension of the Little Picard Theorem follows: If \(f\) is an entire function on a locally convex space which is not constant, then \(f\) attains each complex number, with one possible exception.

Let \(U\) denote a nonvoid open subset of a complex locally convex space \(E\), and let \({\mathcal H}(U)\) be the space of all complex-valued holomorphic functions on \(U\), endowed with the topology of uniform convergence on the compact subsets of \(U\). In Section 2, it is proved that every locally bounded family \({\mathcal F} \subset {\mathcal H}(U)\) is normal if \(E\) is separable. The separability hypothesis is necessary here. Some related results are also given. Section 3 is devoted to the study of normal families of holomorphic functions with exceptional values. It is proved that, for a separable locally convex space \(E\) and for a connected open subset \(U\) of \(E\), a family \({\mathcal F}\) of holomorphic functions on \(U\) with two different exceptional values \(a\) and \(b\) (i.e., the functions in \({\mathcal F}\) omit the values \(a\) and \(b\)) is \(\mathbb{C}_\infty\)-normal (i.e., each sequence in \({\mathcal F}\) either admits a subsequence converging in \({\mathcal H}(U)\) with respect to the compact-open topology or admits a subsequence which diverges to infinity uniformly on each compact subset of \(U\)). An extension of the Little Picard Theorem follows: If \(f\) is an entire function on a locally convex space which is not constant, then \(f\) attains each complex number, with one possible exception.

Reviewer: Klaus Dieter Bierstedt (Paderborn)

##### MSC:

46G20 | Infinite-dimensional holomorphy |

46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |

32A19 | Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) |

32H25 | Picard-type theorems and generalizations for several complex variables |