Approximation properties of pairs of subspaces
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This study is concerned with the approximation properties of pairs. For $\lambda \geq 1$, we prove that given a Banach space $X$ and a closed subspace $Z_{0}$, if the pair $(X,Z_{0})$ has the $\lambda$-bounded approximation property ($\lambda$-BAP), then for every ideal $Z$ containing $Z_{0}$, the pair $(Z,Z_{0})$ has the $\lambda$-BAP; further, if $Z$ is a closed subspace of $X$ and the pair $(X,Z)$ has the $\lambda$-BAP, then for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0}$ such that the pair $(Y,Y\cap Z)$ has the $\lambda$-BAP. We also prove that if $Z$ is a separable closed subspace of $X$, then the pair $(X,Z)$ has the $\lambda$-BAP if and only if for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0} \cup Z$ such that the pair $(Y,Z)$ has the $\lambda$-BAP.
Abstract
This study is concerned with the approximation properties of pairs. For $\lambda \geq 1$, we prove that given a Banach space $X$ and a closed subspace $Z_{0}$, if the pair $(X,Z_{0})$ has the $\lambda$-bounded approximation property ($\lambda$-BAP), then for every ideal $Z$ containing $Z_{0}$, the pair $(Z,Z_{0})$ has the $\lambda$-BAP; further, if $Z$ is a closed subspace of $X$ and the pair $(X,Z)$ has the $\lambda$-BAP, then for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0}$ such that the pair $(Y,Y\cap Z)$ has the $\lambda$-BAP. We also prove that if $Z$ is a separable closed subspace of $X$, then the pair $(X,Z)$ has the $\lambda$-BAP if and only if for every separable subspace $Y_{0}$ of $X$, there exists a separable closed subspace $Y$ containing $Y_{0} \cup Z$ such that the pair $(Y,Z)$ has the $\lambda$-BAP.
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