Orthogonal projection question

[naaa00]

Homework Statement

Hello,

H is a Hilbert space. K is a nonempty, convex, closed subset of H. Prove that the orthogonal projection Pk: H → H, is non-expansive:

ll Pk(x) - Pk(y) ll ≤ ll x - y ll

The Attempt at a Solution

So the length between the Pk's, which is in K (convex) is less than the distance of x and y, (x and y not in K, I assume.)

That's my thought, but I'm getting a bit confused proving it, as I'm mixing things with " ll x - Pk(x) ll ≤ ll x - y ll, " using length of vectors.

Any hint? Thanks.

 

[micromass]

Can you tell us something more about the projection $P_K$? How is this defined for example? What theorems do you think are relevant?