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Thread: Projection On L2 Space

  1. #1
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    Projection On L2 Space

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

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

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

    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
    A Plied Mathematician
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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 $\displaystyle (P^{2}=P)$. Couldn't you multiply each $\displaystyle u$ by the scalar

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

    So,

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