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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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The reason why the function is integrable is because you can bound the integral of $|u|^p$ by $\underset{t \in \mathrm{supp}(u)}{\sup} |u(t)|^p \times \mu (\mathrm{supp}(u))$, where the supremum is taken over the support of $u$ and $\mu(\mathrm{supp}(u))$ is the measure of the support. By compactness of the support and by continuity of $u$ this is finite.

Hope that helps,

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Yes: $|u|^2$ is continuous and compactly supported so $\int |u|^2$ is finite.