zbMATH — the first resource for mathematics

A Montel theorem for holomorphic functions on infinite dimensional spaces that omit the values 0 and 1. (English) Zbl 1172.46030
Consider the family of holomorphic maps from a connected complex Banach manifold into the Riemann sphere, omitting three specified values. The author proves that this family is normal in the sense that every net contains a subnet converging in the compact-open topology to either a member of the family or one of the three constant maps determined by the omitted values.

46G20 Infinite-dimensional holomorphy
32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.)
Full Text: DOI Link
[1] C. Carathéodory, Theory of Functions of a Complex Variable, Vol. II, Chelsea, New York 1960.
[2] S. B. Chae, Holomorphy and Calculus in Normed Spaces, Marcel Dekker, New York 1985. · Zbl 0571.46031
[3] S. Dineen, The Schwarz Lemma, Clarendon Press, Oxford 1989. · Zbl 0708.46046
[4] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer, London 1999 · Zbl 1034.46504
[5] L. A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy, North Holland, Amsterdam, 1979, 345–406.
[6] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc., Providence 1957. · Zbl 0078.10004
[7] J. L. Kelley, General Topology, Van Nostrand, Princeton 1955.
[8] J. Mujica and P. Takatsuka, A Schottky-type theorem for starlike domains in Banach spaces, Proc. Amer. Math. Soc. 135 (2007) no.4, 1141–1144. · Zbl 1124.46024
[9] P. Takatsuka, Normal families of holomorphic functions on infinite dimensional spaces, Port. Math. (N.S.) 63 (2006) no.3, 351–362. · Zbl 1124.46025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.