The space $( \ell^2 ,\lVert \cdot \rVert \_2 )$ is a Hilbert space. The space 
 $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert\_\infty )$ is a Banach space. Does X have fixed point property? (For any closed convex bounded subset $C\subseteq X $ and any nonexpansive map $T:C\to C $ there is a $x\in C$ such that $T(x)=x$)


The space $X$ isn't uniformly convex so I can't use theorems about uniformly convex. There is no theorems about F.P.P in products of spaces and I have no another idea. Does someone have any idea?



Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
I think that the answer is *yes*, and that it should follow from the following facts:


1. every Hilbert space is uniformly convex, hence it has normal structure;
2. the direct sum of two Banach spaces with normal structure, endowed with the infinity norm, has again normal structure (Belluce-Kirk-Steiner, Pacific Journal Math. 1968);
3. the finite direct sum of separable Banach spaces (with any of the possible equivalent norm on it) is itself reflexive, in particular $X = \ell^2 \oplus \ell^2$ is reflexive;
4. normal structure on a reflexive Banach space implies FPP (Kirk, Amer. Math. Monthly 1965).


