Schauder fixed point theorem

Let $X$ be a normed 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 the unit disc of a finite dimensional vector space is always convex and compact hence this theorem extends Brouwer Fixed Point Theorem.

Notice that the space $X$ is not required to be complete, however the subset $K$ being compact,is complete with respect to the metric induced by $X$.