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  1. #1
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    map

    Hey,

    I have a map:

    $\displaystyle T: L^{p}(-2,2) \to L^{p}(-2,2) $, $\displaystyle (Tf)(x)=xf(x)$

    $\displaystyle T$ maps from $\displaystyle L^{p}(-2,2)$ to $\displaystyle L^{p}(-2,2)$ because:

    $\displaystyle \int_{-2}^{2}|(Tf)(x)|^{p}dx=\int_{-2}^{2}|xf(x)|^{p}dx=\int_{-2}^{2}|x|^{p}|f(x)|^{p}dx=?$

    How to continue so that it shows that $\displaystyle \int_{-\infty}^{\infty}|f(x)|^{p}dx < \infty$
    Last edited by surjective; Apr 12th 2011 at 05:42 PM.
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  2. #2
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    $\displaystyle \int_{-2}^{2}|(Tf)(x)|^{p}\,dx=\int_{-2}^{2}|xf(x)|^{p}\,dx=\int_{-2}^{2}|x|^{p}|f(x)|^{p}\,dx \leqslant \int_{-2}^{2}2^{p}|f(x)|^{p}\,dx = \ldots$.
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  3. #3
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    This is because the lrgest value x may assume is 2. Right?
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  4. #4
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    Quote Originally Posted by surjective View Post
    This is because the lrgest value x may assume is 2. Right?
    Yes.
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