# Two fixed point theorems in Banach space

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

Theorem 1 (Brouwer fixed point theorem) Let ${M \subset {\mathbb R}^n}$ be a convex compact set, for any continuous function ${f : M \rightarrow M}$, there exists a point ${x_0 \in M}$ such that ${f (x_0) = x_0}$.

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.