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Post Reopened by Joonas Ilmavirta, Daniel Moskovich, Yemon Choi, Johannes Hahn, Will Sawin
Removed abbreviations, fixed laxout, added tags
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Johannes Hahn
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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 sp, $Y$ a subspace of it and $f$ a $1$-Lip map from $X$ to $\mathbb{R}$. Is it possible to get an extn of $f$ from $X$ to $\mathbb{R}$ with Lip constant $1$?

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 dim. subsp, the resultdimensional subspace follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? Let us assume $X=C(K)$, $K$ is cptcompact, $T_2$  (can assumeassume metrizable alsotoo if you want) and $Y=\{f\in X:f|_D=0\}$ (iei.e. an M-ideal) of $X$. Now can a real valued $1$-LipLipschitz map from $Y$ necessarily hashave a similar extn extension? CanWill it be unique ?

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 sp, $Y$ a subspace of it and $f$ a $1$-Lip map from $X$ to $\mathbb{R}$. Is it possible to get an extn of $f$ from $X$ to $\mathbb{R}$ with Lip constant $1$? The case when $X$ is a metrizable tvs and $Y$ a finite dim. subsp, the result follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? Let us assume $X=C(K)$, $K$ is cpt, $T_2$(can assume metrizable also) and $Y=\{f\in X:f|_D=0\}$ (ie an M-ideal) of $X$. Now can a real valued $1$-Lip map from $Y$ necessarily has a similar extn ? Can it be unique ?

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 ?

a precise problem related to my earlier question is mentioned.
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Tanmoy Paul
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Like Hahn Banach Theorem is it necessaryThe 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 from. I know only about the result by Kirszbrzun and very few of its development. The main problem is: Let $X$ be a real Banach sp, $Y$ a subspace of it and $f$ a locally convex topological vector space has$1$-Lip map from $X$ to $\mathbb{R}$. Is it possible to get an extn of $f$ from $X$ to $\mathbb{R}$ with Lip constant $1$? The case when $X$ is a norm preserving extension overmetrizable tvs and $Y$ a finite dim. subsp, the whole spaceresult follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? By the norm Let us assume $X=C(K)$, $K$ is cpt, $T_2$(can assume metrizable also) and $Y=\{f\in X:f|_D=0\}$ (ie an M-ideal) of $X$. Now can a scalarreal valued Lipschitz$1$-Lip map we mean the number $\inf_{x,y, x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$.from $Y$ necessarily has a similar extn ? Can it be unique ?

Like Hahn Banach Theorem is it necessary that a Lipschitz map from a subspace of a locally convex topological vector space has a norm preserving extension over the whole space ? By the norm of a scalar valued Lipschitz map we mean the number $\inf_{x,y, x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$.

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 sp, $Y$ a subspace of it and $f$ a $1$-Lip map from $X$ to $\mathbb{R}$. Is it possible to get an extn of $f$ from $X$ to $\mathbb{R}$ with Lip constant $1$? The case when $X$ is a metrizable tvs and $Y$ a finite dim. subsp, the result follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? Let us assume $X=C(K)$, $K$ is cpt, $T_2$(can assume metrizable also) and $Y=\{f\in X:f|_D=0\}$ (ie an M-ideal) of $X$. Now can a real valued $1$-Lip map from $Y$ necessarily has a similar extn ? Can it be unique ?

Post Closed as "Needs details or clarity" by Marco Golla, Stefan Kohl, Alex Degtyarev, Bill Johnson, Joonas Ilmavirta
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Tanmoy Paul
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Hahn Banach type extension of a Lipschitz map

Like Hahn Banach Theorem is it necessary that a Lipschitz map from a subspace of a locally convex topological vector space has a norm preserving extension over the whole space ? By the norm of a scalar valued Lipschitz map we mean the number $\inf_{x,y, x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$.