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.

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Geometry Hahn Banach theorem for weak* topology

We have geometric Hahn Banach theorem in standard functional analysis course, saying that there exists a separating hyperplane separates two special sets in usual topology. However this theorem can be used for some special sets in weak* topology. This is the problem 9 in the functional analysis book by Prof. Brezis.

Theorem 1 Let {E} be a Banach space, {A,B \subset E^*} be two nonempty disjoint convex subsets. Assume {A} is open in weak* topology. Then there exist some {x \in E, x \neq 0} and a constant {\alpha} such that the hyperplane {\{ f \in E^* : \langle f, x \rangle_{E^* \times E} = \alpha \}} separates {A} and {B}.

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