# Counterexamples to Brouwer's fixed point theorem for the closed unit ball in Banach space

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point.

Brouwer fixed point theorem applies in particular on the close unit ball of a finite dimensional space.

Do you have counterexample(s) where Brouwer fixed point theorem does not hold on the close unit ball of a Banach space (of infinite dimension)?

Note: this question is a refinement of this one which was put on hold.

• Does the fact that in infinite dimension the close unit ball is not compact ensure the fact that Brouwer fixed point theorem cannot be applied ? In that case there will be a lot of counterexample – Maman Oct 25 '16 at 15:07

A comment in the other question mentioned that the wiki page for the Brouwer Fixed Point Theorem had a counterexample for the Hilbert space $$\ell^2$$. I'll adapt the statement there to $$\ell^p$$ where $$p < \infty$$ so you can compare the two:
In the Banach space $$\ell^p$$ of $$p$$-norm summable real (or complex) sequences, consider the map $$f : \ell^p → \ell^p$$ which sends a sequence $$x_n$$ from the closed unit ball of $$\ell^p$$ to the sequence $$y_n$$ defined by
$$y_0=\sqrt[p]{1-\|x\|_p^p} \,\,\textrm{ and } \,\, y_n=x_{n-1} \textrm{ for } n\geq 1$$
It is not difficult to check that this map is continuous, has its image in the unit sphere of $$\ell^p$$, but does not have a fixed point.
• Thanks. I imagine that you meant $y_0=\sqrt[p]{1-\|x\|_p^p}$ – mathcounterexamples.net Jun 26 '15 at 12:07