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?
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?
Theorem 8.2.2 of Larsen, "Functional Analysis", Marcel Dekker, 1973
This is Corollary 5.2.17 in the book "An introduction to Banach space theory" by Megginson, where the term "uniformly rotund" is used instead of "uniformly convex".
For completeness, I sketch the proof.