TITLE: Hahn Banach type extension of a Lipschitz map
QUESTION [5 upvotes]: The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by Kirszbrzun and very few of its development. The main problem is:

Let $X$ be a real Banach space, $Y$ a subspace of it and $f$ a $1$-Lipschitz map from $Y$ to $\mathbb{R}$. Is it possible to get an extension of $f$ from $X$ to $\mathbb{R}$ with Lipschitz constant $1$?

The case when $X$ is a metrizable tvs  and $Y$ a finite dimensional subspace follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? Let us assume $X=C(K)$, $K$ is compact, $T_2$ (assume metrizable too if you want) and $Y=\{f\in X:f|_D=0\}$ (i.e. an M-ideal) of $X$. Now can a real valued $1$-Lipschitz map from $Y$ necessarily have a similar extension? Will it be unique ?

REPLY [11 votes]: If $X$ is any metric space, $Y$ any subset of $X$, and $f: Y \to \mathbb{R}$ a Lipschitz function, then there is an extension $\tilde{f}: X \to \mathbb{R}$ of $f$ with the same Lipschitz constant. This easy result (just extend the function one point at a time) has been rediscovered many times. See Theorem 1.5.6 of my book Lipschitz Algebras.
The result does not follow from the paper of Johnson and Lindenstrauss that you mention. The case when $X$ is "metrizable" is meaningless as you need an actual metric to have a notion of Lipschitz map. The answer by MKO is basically unrelated to the question as far as I can tell.