# Math Help - Hilbert Spaces

1. ## Hilbert Spaces

Hello,

I need help with this question..

Thank you .

2. Here's a sketch of the proof. Essentially, w_0 is defined as the closest point to u in W.

Let d be the distance from u to W, $d:= \inf\{\|u-w\|:w\in W\}$. We don't yet know that the infimum is attained, but we can choose a sequence (w_n) in W such that $\|u-w_n\|\to d$ as n→∞.

Apply the parallelogram identity $\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2$, with $x = u-w_m,\ y=u-w_n$, to get $4\|u-\tfrac12(w_m+w_n)\|^2 + \|w_m-w_n\|^2 = 2\|u-w_m\|^2 + 2\|u-w_n\|^2$. Since $\tfrac12(w_m+w_n)\in W$, the first term on the left side is ≥d^2, and you should be able to deduce that $\|w_m-w_n\|\to0$. Thus (w_n) is a Cauchy sequence and (by the completeness of V) it converges to a point $w_0\in W$, which is the orthogonal projection of u on W.