On relations between weak approximation properties and their inheritances to subspaces (Q855676)

This is a follow-up paper to [\textit{C.~Choi} and \textit{J.~M.~Kim}, J.~Math.\ Anal.\ Appl.\ 316, No.\,2, 722--735 (2006; Zbl 1098.46016)] and [\textit{J.~M.~Kim}, J.~Math.\ Anal.\ Appl.\ 321, No.\,2, 569--575 (2006; Zbl 1109.46022), see the preceding review], where the authors say that a Banach space \(X\) has the (metric) weak approximation property if every compact operator \(T: X\to X\) is the limit of a net of finite-rank operators (of norm \({\leq\| T\| }\)) for the topology of uniform convergence on compact sets. Here the author shows that a separable dual space with the weak approximation property has the metric weak approximation property; the proof of the classical case of the (metric) approximation property carries over verbatim. It is also shown for a subspace \(Y\) of a Banach space \(X\) with the weak approximation property that, under some additional assumption concerning the approximability of compact operators on \(X^*\) by weak\(^*\) continuous ones, \(Y\) inherits the weak approximation property if \(Y^\bot\) is complemented in \(X^*\).