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 Approximation properties of pairs of subspaces Bull. Korean Math. Soc.Published online 2019 May 16 Keun Young Lee Sejong University Abstract : The paper is concerned with approximation properties of pairs. For $\lambda \geq 1$, we prove that given a Banach space $X$ and closed subspace $Z_{0}$, the pair $(X,Z_{0})$ has the $\lambda$-bounded approximation property ($\lambda$-BAP) if and only if for every ideal $Z$ containing $Z_{0}$, the pair $(Z,Z_{0})$ has the $\lambda$-BAP and that if $Z$ is a closed subspace of $X$ and the pair $(X,Z)$ has the $\lambda$-BAP, then every separable subspace $Y_{0}$ of $X$, there exists a separable subspace $Y$ containing $Y_{0}$ such that the pair $(Y,Y\cap Z)$ has the $\lambda$-BAP. We also provide that if $Z$ be a separable closed subspace of $X$, then the pair $(X,Z)$ has the $\lambda$-BAP if and only if every separable subspace $Y_{0}$ of $X$, there exists a separable subspace $Y$ containing $Y_{0}$ and $Z$ such that the pair $(Y,Z)$ has the $\lambda$-BAP. Keywords : bounded approximation property of pairs, approximation property of pairs, ideals MSC numbers : 46B28; 47L20 Full-Text :