Here is a famous fixed point theorem in finite dimension by Brouwer:

Theorem 1 (Brouwer fixed point theorem)Let be a convex compact set, for any continuous function , there exists a point such that .

There are couple of ways to extend this theorem to Banach spaces. First recall that a mapping between two Banach spaces is called **compact**, if the mapping is continuous (not necessarily linear) and the images of bounded sets are pre-compact.