[[Hilbert space]]
# Nearest point of a convex subset of a Hilbert space

Let $X$ be a [[Hilbert space]] and let $A \sube X$ be a inhabited, [[Topological space|closed]], [[convex subset]].
Then for any $x \in X$ there exists a unique $a \in A$ such that $\|x-a\|=d(x,A)$ #m/thm/anal/fun 
where
$$
\begin{align*}
d(x,A) = \inf \{ \|x-a\|: a \in A \}
\end{align*}
$$

> [!missing]- Proof
> #missing/proof


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