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Let $\Omega:= ]0,2\pi[^d\subset \mathbb R^d$, and let's consider following spaces:

  • $L^2(\mathbb{\Omega}) = L^2(\Omega, \mu_L)$ := space of $\Omega$-valued functions defined on $\Omega$ for which the square of the absolute value is Lebesgue integrable relatively to standart Lebesgue measure.

  • $C_c^{\infty}(\Omega) = \{f\in C^{\infty}(\Omega):\,\,\overline{\mathrm{supp}\,f}\subsetneq \Omega\}$ := space of functions that are both smooth, in the sense of having continuous (strong) derivatives of all orders, and compactly supported.

I'm looking for the proof of : $C_c^{\infty}(\Omega)$ is dense in $L^2(\Omega)$.

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  • $\begingroup$ Hint: Use the fact that simple functions are dense in $L^2$ $\endgroup$
    – Dunham
    Jun 5, 2017 at 10:54

1 Answer 1

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Let $f\in L^2(]0, 2\pi[)$ and take an approximation to identity $\eta_\epsilon \in C_c^\infty(\mathbb R)$ such that $\operatorname{supp} \eta_\epsilon \subset ]-\epsilon, \epsilon[.$ Then let $f_\epsilon = \eta_\epsilon * f \chi_{[2\epsilon, 2\pi-2\epsilon]}$ where $\chi_A(x) = 1$ if $x\in A$ and $=0$ if $x \notin A$. Then $f_\epsilon \in C_c^\infty(\Omega)$ with support $\subset [\epsilon,2\pi-\epsilon]$ and $f_\epsilon \to f$ in $L^2(\Omega)$.

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