具有球覆盖性质的对偶巴拿赫空间
Dual Banach spaces with the ball-covering property
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中文总结 AI 辅助
研究对偶巴拿赫空间球覆盖性质及与预对偶单位球几何联系，给出单位球是切片可数确定集的充要条件等，得到对偶空间一致球覆盖性质的充分条件，应用于算子空间和利普希茨空间，回答相关问题。
中文摘要 AI 辅助
我们研究对偶巴拿赫空间的球覆盖性质及其与预对偶单位球几何结构的联系。主要结果之一表明，对于每个可分巴拿赫空间\(X\)，单位球\(B_X\)是切片可数确定集当且仅当\(\operatorname{bc}(X^*) = 1\)，其中\(\operatorname{bc}(\cdot)\)是由A. J. Guirao、A. Lissitsin和V. Montesinos引入的球覆盖指标。我们得到了对偶空间中一致球覆盖性质的几个充分条件，包括\(K < 2\)时具有\(K -\)无条件基的空间的对偶，以及单位球是一组一致强暴露点的闭凸包的可分空间的对偶。常数\(2\)是精确的：存在一个具有\(2 -\)无条件基的空间，其对偶不具有球覆盖性质。还给出了在算子空间和利普希茨空间中的应用。特别地，对于每个\(1 < p < \infty\)，\(\mathcal L(L_p[0,1])\)具有一致球覆盖性质，这回答了Q. Bao、R. Liu和J. Shen提出的一个问题。作为对利普希茨空间的应用，我们证明当\(M\)是可分完备超度量或赫尔德度量空间时，\(\operatorname{Lip}_0(M)\)具有一致球覆盖性质。
英文摘要
We study ball-covering properties of dual Banach spaces and their connections with the geometry of predual unit balls. One of our main results shows that, for every separable Banach space $X$, the unit ball $B_X$ is a slicely countably determined set if and only if $\operatorname{bc}(X^*)=1$, where $\operatorname{bc}(\cdot)$ is the ball-covering index introduced by A. J. Guirao, A. Lissitsin, and V. Montesinos. We obtain several sufficient conditions for the uniform ball-covering property in dual spaces, including duals of spaces with a $K$-unconditional basis for $K<2$, and duals of separable spaces whose unit ball is the closed convex hull of a set of uniformly strongly exposed points. The constant $2$ is sharp: there is a space with a $2$-unconditional basis whose dual fails the ball-covering property. Applications are given to spaces of operators and to Lipschitz spaces. In particular, $\mathcal L(L_p[0,1])$ has the uniform ball-covering property for every $1<p<\infty$, which answers a question posed by Q. Bao, R. Liu, and J. Shen. As an application to Lipschitz spaces, we prove that $\operatorname{Lip}_0(M)$ has the uniform ball-covering property whenever $M$ is a separable complete ultrametric or Hölder metric space.
