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 ${\mathcal{B}}_{⁎}$, the biduality result that ${\mathcal{B}}_0^{⁎}={\mathcal{B}}_{⁎}$ and ${\mathcal{B}}_{⁎}^{⁎}=\mathcal{B}$, and a formula for the distance from an element $f\in \mathcal{B}$ to ${\mathcal{B}}_0$.

中文翻译：

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原子分解，两个恒星定理以及Bourgain-Brezis-Mironescu空间和其他大空间的距离

给定Banach空间E具有由一组算子集合引起的极值型范数，我们证明E是对偶空间，并提供其前生的原子分解。我们将此结果以及其中一位作者先前获得的结果应用于函数空间$\mathcal{乙}$最近由Bourgain，Brezis和Mironescu介绍。这产生了原子的原子分解${\mathcal{乙}}_{⁎}$，结果是 ${\mathcal{乙}}_0^{⁎}={\mathcal{乙}}_{⁎}$ 和 ${\mathcal{乙}}_{⁎}^{⁎}=\mathcal{乙}$，以及到元素的距离的公式 $F\in \mathcal{乙}$ 至 ${\mathcal{乙}}_0$。