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 $C(Y,E)\to C(X,E)$?

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, the Tietze extension theorem can thus be rephrased saying that $\mathbb{R}$ is an AR(normal) space.

If in the 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 $C(Y,E)\to C(X,E)$?

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?

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 $C(Y,E)\to C(X,E)$?