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Lately, I have been studying Hilbert spaces. I want to know how to prove that continuos functions of compact support are dense subspaces of $L^2(\mathbb R^d)$?

Parth Raghav
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    do it in stages: first, characteristic functions can be approxmated by continuous functions w/compact support: use regularity of the measure and Urysohn Lemma (or a construction from scratch). Then extend the result to simple functions and from there to $\mathscr L^p$ -functions. – Matematleta Nov 03 '16 at 04:09
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    It reduces to showing that the continuous functions are dense in $L^2$, i.e. $\lim_{\epsilon \to 0} |f-f \ast \phi_\epsilon |{L^2} = 0$ where $f \in L^2$ and $\phi\epsilon$ is a sequence of continuous and compactly supported functions approximating the identity of the convolution – reuns Nov 03 '16 at 04:41

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