# Projection over a Hilbert space.

• Dec 24th 2011, 01:08 PM
Ulysses
Projection over a Hilbert space.
Hi there. I have this exercise, and I have no idea on how to work it.

It says: Let Y be a closed subspace from a Hilbert space H, and P the orthogonal projection of H over Y. Demonstrate that:

a) \$\displaystyle P^2=PoP=P\$ which means, P is an idempotent operator.
b) \$\displaystyle P |_Y=Id\$ which means that the P operator restricted to the subspace Y coincides with the identity.

I haven't boarded the problem yet, but when I try it, I'll let you know. I how to read more theory right now, but I'm far from proving this, and perhaps you could help me. I don't even know what the o in PoP means.

Bye there, and thanks in advance.
• Dec 24th 2011, 03:55 PM
girdav
Re: Projection over a Hilbert space.
\$\displaystyle \circ\$ means the composition: \$\displaystyle P\circ P(x)=P(P(x))\$. The properties you have to show are consequences of the fact that \$\displaystyle Px\$ is by definition in \$\displaystyle Y\$.
• Dec 25th 2011, 02:41 PM
Ulysses
Re: Projection over a Hilbert space.
I see now what the exercise is asking (I've just read the theory now). And I think I know how this proprieties are given. I know the projection is unique. Thinking in a point being projected over a line makes both properties pretty clear. Once it's projected, if I take the projection again, I get the same point. And for any point in a line the projection will give the same point, I think that will give the identity, right? anyway, could you help me a bit with the maths and the notation? I think I get it, but I'm not sure on how to say this in a mathematical fashion.