论文标题
通过基本的生物三相系统的平滑和多面体规范
Smooth and polyhedral norms via fundamental biorthogonal systems
论文作者
论文摘要
令$ \ mathcal {x} $为带有基本生物体系系统的Banach空间，让$ \ Mathcal {y} $为系统的向量跨越密集的子空间。我们证明，$ \ Mathcal {y} $接纳了一个$ c^\ infty $ -Smooth Norm，在本地取决于有限的许多坐标（简称LFC），以及当地依赖于有限的许多坐标的多面性规范。结果，我们还证明了$ \ Mathcal {y} $在本地有限，$σ$ - 均匀离散$ c^\ infty $ -smooth和lfc unity和a $ c^1 $ -smmooth lur norm norm。该定理实质上概括了文献中存在的几个结果，并给出了有关这种密集子空间中平滑度的完整图片。 Our result covers, for instance, every WLD Banach space (hence, all reflexive ones), $L_1(μ)$ for every measure $μ$, $\ell_\infty(Γ)$ spaces for every set $Γ$, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann代数。此外，在Martin Maxter {\ sf mm}下，我们的结果涵盖了所有密度$ω_1$的BANACH空间。
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, $σ$-uniformly discrete $C^\infty$-smooth and LFC partitions of unity and a $C^1$-smooth LUR 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 WLD Banach space (hence, all reflexive ones), $L_1(μ)$ for every measure $μ$, $\ell_\infty(Γ)$ spaces for every set $Γ$, $C(K)$ 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 {\sf MM}, all Banach spaces of density $ω_1$ are covered by our result.
