
Hilbert space proof
let F be a closed subspace of a hilbert space H. Let P:H>F be defined by xP(x)<_ xy 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

Re: Hilbert space proof
You can invoke
$\displaystyle \langle xP_F(x),u\rangle=0,\forall u\in F$
to show that
$\displaystyle \langle xP_F(x),x'\rangle=0, x'\in F^{\perp}$ unique.
Defining now $\displaystyle P_{F^{\perp}}(x)=x'$, you will have proven what is required.