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Math Help - Projection On L2 Space

  1. #1
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    Projection On L2 Space

    Let V=L^2(\Omega) and for r>0:

    K=\{v\in V : ||v||_{L^2(\Omega)}\leq r\}.

    For any u\in V, find its projection on K.

    I have no idea where to start. I know that the projection map has certain properties (linear, self-adjoint, operator norm of 1, whatever), but is there any specific formula to find it?
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  2. #2
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    Hmm. Interesting problem. The trick is that applying the projection operator twice should be the same as applying it just once (P^{2}=P). Couldn't you multiply each u by the scalar

    C_{u}=\dfrac{r}{\|u\|_{L^{2}(\Omega)}}?

    So,

    Pu=\dfrac{ur}{\|u\|_{L^{2}(\Omega)}}.
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