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Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support.

Is $u$ in $L^2(\Bbb{R}^d)$?

I think that $u$ is in $L^2(\Bbb{R}^d)$ since $u$ has compact support.

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    Any bounded measurable function $u$ with compact support is integrable. Since its any positive power $|u|^p$ also fulfill these conditions, $u \in L^p$ for any $p > 0$.2012-08-22

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