Consider the following subspace of $L^1(0,1)$. Let $Z$ be the subspace of all integrable functions that integrate to zero. So, $f \in Z$ implies $$ \int_0^1 f(x) \, dx = 0.$$
How to show that this space is not L-embedded (or is it L-embedded)?
It is known that $L^1(0,1)$ is L-embedded, but this property does not always pass to subspaces.