# Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

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?

• Maybe use (somehow) the fact that the norm $\|\cdot\|_\infty$ is the limit of uniformly convex norms $\|\cdot\|_p$ for $p\to\infty$? Commented Nov 12, 2018 at 8:43
• No. this is not solution. I tried for ten days but i didn't achieve solution Commented Nov 12, 2018 at 8:46
• How is your norm defined? Is $$|| \,\cdot \,||_{\infty} = \max \left( || \,\cdot \,||_2, \; || \,\cdot \,||_2 \right)?$$ Commented Nov 12, 2018 at 9:34
• Yes . .the norm is this Commented Nov 12, 2018 at 9:36

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).

• But normal structure implies W.F.P.P And I didn't see number (3) somewhere. Commented Nov 12, 2018 at 10:13
• Wouldn't WFPP and FPP coincide on reflexive spaces? Commented Nov 12, 2018 at 21:09
• Yes. But X maybe isn't reflexive. Commented Nov 13, 2018 at 2:12
• The finite direct sum of reflexive Banach spaces is itself reflexive. Note also that your norm is equivalent to the (finite) $L^p$-norm $$||(x, \, y)||_p = \left( ||x||^p + ||y||^p \right)^{1/p},$$ see en.wikipedia.org/wiki/Banach_space#Reflexivity Commented Nov 13, 2018 at 6:48