Let $\mathcal {X}$ be a Banach space with a fundamental biorthogonal system, and let $\mathcal {Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal {Y}$ admits a $C^\infty $-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal {Y}$ admits locally finite, $\sigma $-uniformly discrete $C^\infty $-smooth and LFC partitions of unity and a $C^1$-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), $L_1(\mu )$ for every measure $\mu $, $\ell _\infty (\Gamma )$ spaces for every set $\Gamma $, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density $\omega _1$ are covered by our result.

中文翻译：

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通过基本双正交系统的平滑和多面体范数

令$\mathcal {X}$ 为具有基本双正交系统的Banach 空间，令$\mathcal {Y}$ 为系统向量跨越的稠密子空间。我们证明$\mathcal {Y}$ 承认一个局部依赖于有限多个坐标的$C^\infty $-平滑范数（简称LFC），以及一个局部依赖于有限多个坐标的多面体范数。因此，我们还证明了 $\mathcal {Y}$ 承认局部有限、$\sigma $-均匀离散的 $C^\infty $-smooth 和 LFC 统一分区以及 $C^1$-局部均匀平滑圆规。这个定理基本上概括了文献中存在的几个结果，并给出了关于这种密集子空间中的平滑度的完整图片。例如，我们的结果涵盖了每个弱 Lindelöf 确定的 Banach 空间（因此，所有自反空间），每个度量的 $L_1(\mu )$ $\mu $, $\ell _\infty (\Gamma )$ 每个集合的空间 $\Gamma $, $C(K)$ 个空间，其中 $K$ 是 Valdivia compactum或紧致阿贝尔群，Asplund 空间的对偶，或冯诺依曼代数的预对偶。此外，在 Martin 最大 MM 下，我们的结果涵盖了密度为 $\omega _1$ 的所有 Banach 空间。