# Math Help - map

1. ## map

Hey,

I have a map:

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

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

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

How to continue so that it shows that $\int_{-\infty}^{\infty}|f(x)|^{p}dx < \infty$

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

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

4. Originally Posted by surjective
This is because the lrgest value x may assume is 2. Right?
Yes.