论文标题
$ \ ell_p（γ）$和操作员范围的密集子空间中的平滑规范
Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges
论文作者
论文摘要
For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(Γ)$ comprising all elements $y$ such that $y \in \ell_q(Γ)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on finitely many坐标。此外，可以选择这样的规范以近似$ \ weled \ vert \ cdot \ right \ vert_p $ -norm。这提供了具有平滑标准的$ \ ell_p（γ）$的密集子空间的示例，具有最大可能的线性维度，并且不能作为生物息肉系统的线性跨度获得。此外，当$ p> 1 $或$γ$可数时，此类子空间还包含密集的操作员范围；另一方面，$ \ ell_1（γ）$中没有不可分割的操作员范围允许$ c^1 $ -SMOOTH NORM。
For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(Γ)$ comprising all elements $y$ such that $y \in \ell_q(Γ)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $\left\Vert\cdot \right\Vert_p $-norm. This provides examples of dense subspaces of $\ell_p(Γ)$ with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $p>1$ or $Γ$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(Γ)$ admits a $C^1$-smooth norm.
