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Given a Banach space $E$ with a supremum-type norm induced by a collection ofoperators, we prove that $E$ is a dual space and provide an atomicdecomposition of its predual. We apply this result, and some results obtainedpreviously by one of the authors, to the function space $\mathcal{B}$introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomicdecomposition 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 $\mathcal{B}_0$.

Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces