My question is: a Banach space $X$ has the approximation property if and only if for every $\epsilon>0$ and every compact subset $K\subset X$, there exists a Lipschitz map $T: X\rightarrow X$ with relatively compact image $T(X)$ such that $\Txx\<\epsilon$, for all $x\in K$. The purpose of my question is to know about metric characterizations of approximation property. Is there any reference on metric characterizations of approximation property?
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1$\begingroup$ I don't think anything is known other than Theorem 5.3 in Godefroy, G.; Kalton, N. J. Lipschitzfree Banach spaces. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. Studia Math. 159 (2003), no. 1, 121–141. $\endgroup$ – Bill Johnson Feb 13 '15 at 23:04

$\begingroup$ Do you think the proof of Theorem 5.3 in Godefroy and Kalton's paper works for the AP? $\endgroup$ – Dongyang Chen Feb 14 '15 at 14:14

$\begingroup$ Do you think my question is interesting or not? $\endgroup$ – Dongyang Chen Feb 14 '15 at 16:13

$\begingroup$ Sure, it is interesting. $\endgroup$ – Bill Johnson Feb 15 '15 at 15:02