Tychonoff fixed point theorem

Let $X$ be a locally convex topological vector space, and let $K\subset X$ be a non-empty, compact, and convex set.Then given any continuous mapping $f:K\to K$there exists $x\in K$ such that $f(x)=x$.

Notice that a normed vector space is a locally convex topologicalvector space so this theorem extends the Schauder fixed point theorem.