The open Problem 1.e.9 in Lindenstrauss-Tzafriri states:
Let $X$ be a Banach space such that every compact $T:X\to X$ is a limit in norm of finite rank operators from $X$ into itself. Does $X$ have the A.P.?
I understand how this statement is stronger than the implication (v)$\Rightarrow$ (i) in Theorem 1.e.4, however I had the impression for a long time that 1.e.9 is the definition of A.P., that is the norm closure of finite rank operators are the compact operators. What am I missing here?