Nearest point projection in uniformly convex Banach spaces 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?
 A: Theorem 8.2.2 of Larsen, "Functional Analysis", Marcel Dekker, 1973
A: 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. 


*

*We may assume $x=0$, by translation. 

*Let $d = \|y_n\|\to\inf \{\|y\| : y\in C\}$. We may replace $C$ with $\{y\in C: \|y\|\le 2d\}$ without changing the problem. This makes $C$ bounded.

*Take a sequence $y_n $ such that $\|y_n\|\to d$.

*Uniformly convex spaces are reflexive. Therefore, $C$ is weakly compact by Alaoglu's theorem. 

*Replace $\{y_n\}$ by a weakly convergent subsequence, and let $y$ be its limit. Note that $y\in C$ because closed convex sets are weakly closed. Hence $\|y\|\ge d$. But since closed  balls around $0$ are also weakly closed, we have $\|y\|\le d$. Thus $\|y\|=d$. This proves the existence. 

*For the uniqueness, strict convexity is enough: the midpoint of a line segment between two "nearest" points would be even closer to $0$.

