Banach fixed point theorem in $\mathbb R^n$

Question

I have a question on Banach fixed-point theorem. It supposes we are in a closed set $C \subset \mathbb R^n$ with the image of $C$ by a function is included in $C : f(C)\subset C$ and we also suppose that $\exists \rho \in ]0, 1[ \text{ such that } \forall v, w \in C \lVert f(v) − f(w)\rVert \leq \rho \lVert v − w \rVert $.

Then $\exists! y \in C \text{ such that } f(y)=y. $

Can $C$ be equal to $\mathbb R^n$ ? Does the result still holds ?

Answer 1

Yes, $C$ can be any complete metric space, so a closed set of $\Bbb R^n$ works just as well as $\Bbb R^n$ itself.