Orthogonal projection question

1. Nov 7, 2012

naaa00

1. The problem statement, all variables and given/known data

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

3. 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 lenght of vectors.

Any hint? Thanks.

2. Nov 7, 2012

micromass

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