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Math Help - Projection over a Hilbert space.

  1. #1
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    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) P^2=PoP=P which means, P is an idempotent operator.
    b) 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.
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  2. #2
    Super Member girdav's Avatar
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    Re: Projection over a Hilbert space.

    \circ means the composition: P\circ P(x)=P(P(x)). The properties you have to show are consequences of the fact that Px is by definition in Y.
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  3. #3
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    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.
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