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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On compact and bounding holomorphic mappings

Author: Mikael Lindström
Journal: Proc. Amer. Math. Soc. 105 (1989), 356-361
MSC: Primary 46G20; Secondary 58C10
MathSciNet review: 933517
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Abstract: Let $ E$ and $ F$ be complex Banach spaces. We say that a holomorphic mapping $ f$ from $ E$ into $ F$ is compact respectively bounding if $ f$ maps some neighbourhood of every point of $ E$ into a relatively compact respectively bounding subset of $ F$. Recall that a subset of $ E$ is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on $ E$. Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.

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Article copyright: © Copyright 1989 American Mathematical Society