THREE-SPACE PROBLEM FOR SOME APPROXIMATION PROPERTIES

Abstract

Suppose that $M$ is a closed subspace of a Banach space $X$ such that $M^{\bot}$ is complemented in the dual space $X^{*}$, where $M^{\perp} = \{x^{*} \in X^{*}: x^{*}(m) = 0$ for all $m \in M\}$. Godefroy and Saphar [4] study the three-space problem for the approximation properties on $(X,M)$. In this paper, we extend some of their results and solve the three-space problem for the weak bounded approximation property on $(X,M)$, which was introduced in Lima and Oja [10].