2003-5.md

course	course_year	question_number	tags	title	year
Analysis II	IB	5	IB 2003 Analysis II	$3 . \mathrm{II} . 11 \mathrm{~F} \quad$	2003

State and prove the Contraction Mapping Theorem.

Let $(X, d)$ be a bounded metric space, and let $F$ denote the set of all continuous maps $X \rightarrow X$. Let $\rho: F \times F \rightarrow \mathbb{R}$ be the function

$$\rho(f, g)=\sup {d(f(x), g(x)): x \in X}$$

Show that $\rho$ is a metric on $F$, and that $(F, \rho)$ is complete if $(X, d)$ is complete. [You may assume that a uniform limit of continuous functions is continuous.]

Now suppose that $(X, d)$ is complete. Let $C \subseteq F$ be the set of contraction mappings, and let $\theta: C \rightarrow X$ be the function which sends a contraction mapping to its unique fixed point. Show that $\theta$ is continuous. [Hint: fix $f \in C$ and consider $d(\theta(g), f(\theta(g)))$, where $g \in C$ is close to $f$.]