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Let Y be a closed subspace of a normed linear space X. Show that Y* is isometrically isomorhpic with $X^*/Y^\perp$, where $Y^\perp$ is the set of functionals $\ell$ that vanish on Y.

I have a little problem understanding what I am supposed to do. Do I need to find an bijection between the sets that is group and distance preserving?

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You have to find a linear bijective $X^*/Y^\bot \to Y^*$ that is distance preserving. Note that linear maps $X^*/Y^\bot \to Y^*$ correspond to linear maps $X^* \to Y^*$, which's kernel contains $Y^\bot$. There is a very simple map $X^* \to Y^*$ (keep in mind that $Y \subseteq X$) that will work here.

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    exactly.${}{}{}$2012-11-26