# Hilbert space proof

• April 11th 2013, 08:43 AM
Plato13
Hilbert space proof
let F be a closed subspace of a hilbert space H. Let P:H->F be defined by ||x-P(x)||<_ ||x-y|| for every y in F. In other words P(x) is orthogonal projection of x on F.

Show that orthogonal projection of x on the orthogonal complement of F is = to the identity operator - orthogonal projection of x on F.

Thanks
• January 22nd 2015, 06:44 PM
Rebesques
Re: Hilbert space proof
You can invoke

$\langle x-P_F(x),u\rangle=0,\forall u\in F$

to show that

$\langle x-P_F(x),x'\rangle=0, x'\in F^{\perp}$ unique.

Defining now $P_{F^{\perp}}(x)=x'$, you will have proven what is required.