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In this paper, we study the bounded approximation property for the weighted space
(U) of holomorphic mappings defined on a balanced open subset U of a Banach space E and its predual
is a countable family of weights. After obtaining an
-absolute decomposition for the space
(U), we show that E has the bounded approximation property if and only if
(U) has. In case
consists of a single weight v, an analogous characterization for the metric approximation property for a Banach space E has been obtained in terms of the metric approximation property for the space
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