Let $E$ be a normed vector Space Let $T: E \to E $ an affine map. Let $C$ be a convex compact non empty of $E$ such that $T(C) \subseteq C$. Show that $T$ fixes a point of $C$
I am stucked on this problem because $T$ is not necessarily continuous so I can’t use any of Brouwer/Shauder’s theorem
I tried to build a sequence $(x_n)$ such that $T(x_n)=x_{n+1}$ and to show that it converges but I didn’t succeed.
