The intersection $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ is (allegedly) a Banach space for the norm $\|f\| = \|f\|_1 + \|f\|_2$. Is it also true that $C_c(\mathbb{R})$ is dense with respect to this norm?
I think it's reasonably clear that simple functions are dense... It's enough to show you can approximate any nonnegative $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$. But, for such $f$, it's not difficult to produce a monotone increasing sequence of simple functions $s_n$ converging pointwise to $f$ from below, and then check $\|f-s_n\|_1 \to 0$ and $\|f-s_n\|_2 \to 0$.
OK, so this means we just need to prove that the simple functions are approximable by compactly supported functions in the norm $\| \cdot \|$. Thus, it is enough to take a measureable set $E$ with finite measure and show its characteristic function $\chi_E$ can be approximated. So, I guess, my question reduces to:
Question: If $E \subset \mathbb{R}$ is measureable with finite Lebesgue measure, and $\epsilon > 0$, is there always an $f \in C_c(\mathbb{R})$ such that $\| f - \chi_E\|_1 < \epsilon$ and $\|f - \chi_E\|_2 < \epsilon$?
This holds for any topological space $X$ which is locally compact and Hausdorff. It holds in particular for $\mathbb R$.
Since the Lebesgue measure $\ell$ is regular (and since $\ell(E)<+\infty$), there exist a compact set $K\subset\mathbb R$ and an open set $U\subset\mathbb R$ such that $K\subset E\subset U$ and \begin{equation*} \ell\left(U\setminus K\right)<\frac\varepsilon{2}. \end{equation*} Choose $f\in C_c(\mathbb R)$ according to Urysohn's lemma, such that $\chi_K\le f\le\chi_U$. Then, by Minkowski's inequality, for any $p\in[1,2]$, \begin{align*} \|\chi_{E}-f\|_p &\le\|f-\chi_{K}\|_p+\|\chi_{E}-\chi_{K}\|_p\\ &\le\left(\int\chi_{U\setminus K}(x)\,\mathrm dx\right)^{1/p}+\left(\int\chi_{E\setminus K}(x)\,\mathrm dx\right)^{1/p}\\ &\le2\left(\frac\varepsilon{2}\right)^{1/p}\\ &\le\varepsilon. \end{align*}