# 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.

# Fredholm Alternative and Riesz Schauder Theorem

Theorem (Analytic Fredholm Alternative) Let ${D}$ be a connected open subset of ${{\mathbb C}}$ and ${\mathcal{H}}$ be a separable Hilbert space. Suppose that ${f : D \longrightarrow \mathcal{L} (\mathcal{H})}$ is an analytic operator-valued function such that ${f(z)}$ is compact ${\forall z \in D}$. Then either

1. ${(I - f(z)) ^{-1}}$ does not exist ${\forall z \in D}$.
2. ${(I - f(z)) ^{-1}}$ exists ${\forall z \in D\setminus S}$, where ${S}$ is discrete.

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 ${B}$.