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:



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> 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 $1$?
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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 ?



Hahn Banach type extension of a Lipschitz map
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.


