3.II.11 F3 . \mathrm{II} . 11 \mathrm{~F} \quad

Analysis II | Part IB, 2003

State and prove the Contraction Mapping Theorem.

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

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

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

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

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