In literature I am reading right now, it says $BV(\Omega)$, where $\Omega \subset \mathbb{R}^n$ an open bounded set, is the dual of a separable space. Is it a dual separable Banach space, or not a Banach space at all?
I am asking, because I want to use the weak-* compactness theorem Banach Alaoglu on this space, but then $BV(\Omega)$ needs to be the dual of a separable Banach space.