Assume we have two Banach spaces $X,Y$ over $\mathbb R$ such that $X' \subset Y'$ is a dense subset (with respect to the norm of $Y'$). My question is: Is $Y \subset X$ a dense subset (with respect to the norm of $X$)?
Here, $X'$ denotes the dual space of $X$, i.e. all continuous linear functionals $T:X \longrightarrow \mathbb R$.
I am particularly interested in the example where $X' = W^{1,\infty}(\Omega)$ and $Y = L^1(\Omega)$ (so $Y' = L^\infty(\Omega)$).
As far as I know, there is no explicit characterisation of $X$.
Thank you guys very much in advance.