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Let $Y$ be a subspace of a Banach space $X$.

How can I show that there exists an 1-1 isometric map $\phi\colon Y^*\to X^*$ (need not be linear)?

Also, how can I show that $X^*|_{Y}=Y^*$? Here, $X^*|_{Y}$ is the set of restriction to $Y$ of all $f\in X^*$.

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up vote 2 down vote accepted

Hint: Hahn-Banach might be useful here.

Actually, I think it suffices if $X$ is a normed space, and $Y$ is a linear subspace.

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