  Given a Banach space $E$ with a supremum-type norm induced by a collection of
operators, we prove that $E$ is a dual space and provide an atomic
decomposition of its predual. We apply this result, and some results obtained
previously by one of the authors, to the function space $\mathcal{B}$
introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic
decomposition of the predual $\mathcal{B}_\ast$, the biduality result that
$\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast =
\mathcal{B}$, and a formula for the distance from an element $f \in
\mathcal{B}$ to $\mathcal{B}_0$.
