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Journal of Convex Analysis 18 (2011), No. 4, 1065--1074
Copyright Heldermann Verlag 2011

The Biduality Problem and M-Ideals in Weighted Spaces of Holomorphic Functions

Christopher Boyd
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Pilar Rueda
Dep. de Análisis Matemático, Facultad de Matemáticas, Universidad de Valencia, 46100 Burjasot, Valencia, Spain


Given a weight $v$ on an open subset $U$ of ${\bf C}^n$, ${\cal H}_v(U)$ (resp. ${\cal H}_{v_o}(U)$) denotes the Banach space of holomorphic functions $f$ on $U$ such that $vf$ is bounded on $U$ (resp. converges to $0$ on the boundary of $U$). We show that ${\cal H}_v(U)$ is canonically isometrically isomorphic to the bidual of ${\cal H}_{v_o}(U)$ if and only if ${\cal H}_{v_o}(U)$ is an M-ideal in ${\cal H}_v(U)$ and the associated weights $\tilde v_o$ and $\tilde v$ coincide.

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