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!
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 $C$. $\Box$