Bounded holomorphic mappings and the compact approximation property in Banach spaces

Erhan Çal\iskan

Abstract: We study the compact approximation property in connection with the space of bounded holomorphic mappings on a Banach space. When $U$ is a bounded balanced open subset of a Banach space $E$, we show that the predual of the space of the bounded holomorphic functions on $U$, $G^{\infty}(U)$, has the compact approximation property if and only if $E$ has the compact approximation property. We also show that $E$ has the compact approximation property if and only if each continuous Banach-valued polynomial on $E$ can be uniformly approximated on compact sets by polynomials which are weakly continuous on bounded sets.

Keywords: Banach spaces; compact approximation property; bounded holomorphic functions.

Classification (MSC2000): 46G20, 46B28, 46G25.

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