$L_1 \cap L_2$ is dense in $L_2$? We were talking about Fourier series the other day and my professor said that the requirement that a function be in $L_1 \cap L_2$ wasn't a huge obstacle, because this is dense in $L_2$. Why is this true?

An element $f$ of $\mathbb L^2$ can be approximated for the $\mathbb L^2$ norm by a linear combination of characteristic function of measurable sets of finite measure. Such a function is integrable, hence the function $f$ can be approximated for the $\mathbb L^2$ norm by an element of $\mathbb L^1\cap \mathbb L^2$.
