Problem: Let $C$ be a compact and convex set in a $B^\ast$ space $\mathcal{X}$. Suppose that a mapping $T: C\to C$ is continuous. Show that $T$ has a fixed point in $C$.

I can find the proof for the statement if $\mathcal{X}$ is $\mathbb{R}^n$ on the internet. In that case, every compact convex subset is homeomorphic to a closed unit ball, and we can prove that the statement is true for $C$ is a closed unit ball.

However, I cannot find anywhere if this statement is true for a general $B^\ast$ space $\mathcal{X}$ (this space can be infinite?). I tried to duplicate the proof of the $\mathbb{R}^n$ case, but I cannot describe the function that maps $C$ to $\partial C$. Is there any idea to prove this?

  • 2
    $\begingroup$ This is called Schauder fixed point theorem or Schauder-Tychonoff fixed point theorem. Google the name for more details. $\endgroup$
    – Salcio
    Apr 9, 2023 at 23:01


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