Be $H$ a Hilbert space,$x_{0} \in H$ and $M \subset H$ a closed space. Show that
$\displaystyle\inf_{x\in M} \{ \lVert x-x_0\rVert\} =\sup\{ \lvert\langle x_{0},y\rangle\rvert \;:\; y\in M^{\perp},\lVert y\rVert=1 \}$,
where $\lVert\cdot\rVert$ is the norm of the inner product,
I try use the minimum vector (the problem follow the conditions) in the left of the equality, but I can't work in the right side, any help is appreciated.