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?

  • $\begingroup$ Maybe use (somehow) the fact that the norm $\|\cdot\|_\infty$ is the limit of uniformly convex norms $\|\cdot\|_p$ for $p\to\infty$? $\endgroup$ – Taras Banakh Nov 12 '18 at 8:43
  • $\begingroup$ No. this is not solution. I tried for ten days but i didn't achieve solution $\endgroup$ – Darman Nov 12 '18 at 8:46
  • $\begingroup$ How is your norm defined? Is $$|| \,\cdot \,||_{\infty} = \max \left( || \,\cdot \,||_2, \; || \,\cdot \,||_2 \right)?$$ $\endgroup$ – Francesco Polizzi Nov 12 '18 at 9:34
  • $\begingroup$ Yes . .the norm is this $\endgroup$ – Darman Nov 12 '18 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).

  • $\begingroup$ But normal structure implies W.F.P.P And I didn't see number (3) somewhere. $\endgroup$ – Darman Nov 12 '18 at 10:13
  • 2
    $\begingroup$ Wouldn't WFPP and FPP coincide on reflexive spaces? $\endgroup$ – Dirk Werner Nov 12 '18 at 21:09
  • $\begingroup$ Yes. But X maybe isn't reflexive. $\endgroup$ – Darman Nov 13 '18 at 2:12
  • 2
    $\begingroup$ 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 $\endgroup$ – Francesco Polizzi Nov 13 '18 at 6:48

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