Banach intersection of L1 and L2

The intersection $L^1\cap L^2$ is Banach for $\Vert f{\Vert }_1+\Vert f{\Vert }_2$, but it is generally incomplete when equipped with the $L^1$ norm alone. Truncations of an $L^1$ function outside $L^2$ provide an $L^1$-Cauchy sequence with no limit in the intersection.