approximation property In I. Namioka and R. R. Phelps's your paper "Tensor products of compact convex sets" Pacific Journal of Mathematics, Vol. 31, No. 2, 1969), they gave the following definition of approximation property:
DEFINITION. A Banach space E is said to have the approximation property [2] if for each compact convex subset C of E and each ε > 0, there exists a continuous linear transformation (or equivalently, affine transformation) T: E —> E such that T(E) is finite dimensional and || Tx — x || < ε if $x\in C$. It remains open whether every Banach space has the approximation property.
We know that a linear transformation must be affine. But an affine transformation needn't
be linear. Why did they say "linear transformation (or equivalently, affine transformation) "? Can we get a linear transformation S: E —> E such that S(E) is finite dimensional and
 || Sx — x || < bε if $x\in C$ for some b>0 from an affine transformation T: E —> E such that T(E) is finite dimensional and || Tx — x || < ε if $x\in C$? Why? How can we get it? 
Thanks!
 A: Note: this is no longer open. Per Enflo gave a counterexample in a 1973 Acta paper, and received a live goose from Mazur for that. 
Assume the affine version holds. We can show that the linear version holds in two steps.
1 - The linear version holds on compact convex sets containing $0$.
Proof: let $C$ be such a compact set. Take $\epsilon>0$. We get $T$ affine continuous with finite rank such that $\|Tx-x\|\leq \frac{\epsilon}{2}$ for every $x\in C$. In particular, for $x=0$, $\|T0\|\leq \frac{\epsilon}{2}$. Whence $\|Sx-x\|\leq \epsilon$ on $C$ for the linear map $Sx=Tx-T0$. $\Box$.
2 - The linear version holds on arbitrary compact convex sets.
Proof: let $C$ be any compact convex set and $\epsilon>0$. Then the map $[0,1]\times C\longrightarrow E$ sending $(t,c)$ to $tc$ is continuous, whence its range $C'$ is a compact convex set containing $C$ and $0$. It only remains to apply 1 to $C'$. Note that $C'$ is indeed convex because, when $t_1t_2\neq 0$, we can write $(1-s)t_1c_1+st_2c_2=t_0c_0$ with $t_0=(1-s)t_1+st_2\neq 0$ and $c_0\in C$ by convexity of $C$.
$\Box$
