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Let $X$ be a uniformly convex Banach space, $x\in X$ and $C\subset X$ closed and convex, then there is a unique $y\in C$ with $$\lVert x-y\rVert=\inf_{z\in C}\lVert x-z \rVert.$$

Is there a good book where I can find a proof of this theorem?

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Theorem 8.2.2 of Larsen, "Functional Analysis", Marcel Dekker, 1973

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