Lebesgue integration for $u \in C^{\infty}_c$

Question

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.

Answer 1

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

Answer 2

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,