# Banach Fix Point Theorem

Banach fix point theorem is a very powerful tool in mathematics, in particular, in functional analysis and in operator theory. I was studying for functional analysis earlier this weekend and came across it once again, even though I did not see it coming. We will first look at a few definitions, and then the theorem itself.

Let $(X,d)$ be a metric space. A map $T:X \to X$ is called a contraction if  $d(T(x),T(y)) \le d(x,y)$; it is called a strict contraction the inequality is strict, or equivalently, if there exists $k \in [0,1)$ such that $d(T(x),T(y)) = k d(x,y)$.

As the name suggest, Banach fixed point theorem is a tool to find fix points for contraction mappings in complete metric space, and hence Banach spaces. The statement of the theorem is as follows:

Suppose $(X,d)$ is a complete metric space. If $T:X \to X$ is a strict contraction, then there exists a unique $x^* \in X$ such that $T(x^*) = x^*$. Furthermore, for any $x_0 \in X$, if we set $x_n = T(x_{n-1})$ for all $n \in \mathbb{N}$, then $\lim_{n \in \mathbb{N}} x_n = x^*$.

The proof on Wikipedia is pretty concise, so we will not talk about it here.  It is often used in differential equations to find fixed points of the equation itself. There is yet another clever application that I encounter earlier this weekend, it is used in a theorem by Stampacchia in functional analysis. Maybe we will talk about it next time.