Extension of a continuous function $f:X\rightarrow X$ to a function $g: E\rightarrow E$ such that $X$ is embedded in $E$, a Banach space.

Question

We consider a compact metric space $X$ and a continuous function $f:X\rightarrow X$. I know it's possible to embed $X$ into a Banach or separable Hilbert space $E$ using the weak star topology. However I'm interested in whether it's possible to extend a continuous function such as $f$ to the whole space $E$ once you embed $X$ on $E$. I know something like this should be possible because of representations of dynamical systems on Banach spaces, however I haven't seen a direct proof of what I'm asking here, so any help or source would be greatly appreciated.

Answer 1

As I said in the comment, all what you need is a generalization of the Tietze Extension Theorem due to Dugundji, see

Dugundji, James, An extension of Tietze’s theorem, Pac. J. Math. 1, 353-367 (1951). ZBL0043.38105.

or, for a textbook treatment, Chapter III, Theorem 7.1 in

Borsuk, Karol, Theory of retracts, Monografie Matematyczne. 44. Warszawa: PWN - Polish Scientific Publishers. 251 p. (1967). ZBL0153.52905.

or simply read the proof in the linked Wikipedia article if your French is sufficiently good.

Theorem. Let $A$ be a closed subset of a metric space $X$, let $V$ be a locally convex topological vector space (e.g. any Banach space). Then for every continuous map $f: A\to V$ there exists a continuous extension $F: X\to V$. (Moreover, the image of $F$ can be taken to lie in the closed convex hull of $f(A)$ in $V$, but we will not need this.)

Now, if you have a compact metrizable space $K\subset V$ embedded in a Banach space $V$, and a continuous self-map $f: K\to K$, you simply take $X=V$, $A=K$. Then apply the above extension theorem. (I assume that you know how to prove that $K$ is closed in $V$.)