The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen as a property of the target space $\mathbb{R}$, this leads to the important notion of absolute neighborhood retract, or AR(normal), in Dugundji's notation; Tietze Extension Theorem can thus be rephrased saying that $\mathbb{R}$ is an AR(normal) space.
If in Tietze theorem we restrict the class of domains from normal to metric spaces, by the Dugundji Extension theorem, at least all locally convex topological vector spaces are suitable codomains: any continuous LCTVS-valued function on a closed subset of a metric space can be extended to a continuous function on the whole space.
Of course, this situation in principle allows a wide variety of intermediate situations. The first natural questions, that I would be glad to learn an answer of, are:
Q1. Does Dugundji's theorem hold true for normal spaces, namely, can any continuous LCTVS-valued function on a closed subset of a
normal topological space be extended to a continuous function on the
whole space?
I guess the answer is no, but I can't imagine a counterexample. In case of a negative (or not known) answer:
Q2. Are Banach spaces absolute retract for Hausdorff compact spaces, namely, can any continuous Banach-valued function on a closed subset of a
Hausdorff compact space be extended to a continuous function on the
whole space?
edit After Bill Johnson's answer to question 2, and the other useful comments, I would like to focus on the following question, that should have some good reference in the (wide) literature.
Q3. Let $X$ be a Hausdorff compact topological space, $Y\subset X$ a closed set, $E$ a Banach space. Does there always exist a bounded
linear extension operator $C(Y,E)\to C(X,E)$?
Best Answer
Here is an answer to Q2.
Since compact subsets of Banach spaces are separable, WLOG the target Banach space is separable.
Since all separable infinite dimensional Banach spaces are homeomorphic, WLOG the target Banach space is $c_0$.
Since $c_0$ is a Lipschitz retract of $\ell_\infty$, WLOG the target Banach space is $\ell_\infty$.
The space $\ell_\infty$ clearly has the desired property.
Sorry, Pietro; this being the day after April 1, I could not resist giving this answer.