Continuous Functions with Compact Support Dense in Lebesgue P-Space

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Theorem

Let $C_c \left({\R^n}\right)$ be the space of continuous functions with compact support on $\R^n$.

Let $p \in \R$, $p \ge 1$, and let $\mathcal L^p \left({\lambda^n}\right)$ be Lebesgue $p$-space for Lebesgue measure $\lambda^n$.


Then $C_c \left({\R^n}\right)$ is everywhere dense in $\mathcal L^p \left({\lambda^n}\right)$ with respect to the $p$-seminorm $\left\Vert{\, \cdot \,}\right\Vert_p$.


Proof


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