The celebrated Tietze extension theorem asserts that any continuous real-valued function 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?

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?