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Orthogonal projection question

  1. Nov 7, 2012 #1
    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. jcsd
  3. Nov 7, 2012 #2

    micromass

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