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 $f\colon K \to K$ there exists $x\in K$ such that $f(x)=x$ .

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

Bibliography

1

Rudin, Functional Analysis, Chapter 5.