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 $\displaystyle \int_{-\infty}^{\infty}|f(x)|^{p}dx < \infty$

$\displaystyle \int_{-2}^{2}|(Tf)(x)|^{p}\,dx=\int_{-2}^{2}|xf(x)|^{p}\,dx=\int_{-2}^{2}|x|^{p}|f(x)|^{p}\,dx \leqslant \int_{-2}^{2}2^{p}|f(x)|^{p}\,dx = \ldots$.

This is because the lrgest value x may assume is 2. Right?

Quote:

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Originally Posted by surjective

This is because the lrgest value x may assume is 2. Right?

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Yes. (Nod)