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?
2026-04-12 00:00:10.1775952010
$L_1 \cap L_2$ is dense in $L_2$?
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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 $\mathbb L^1\cap \mathbb L^2$.