在单位球中具有大的弱开子集的强正则巴拿赫空间
Strongly regular Banach spaces with big weakly open subsets in the unit ball
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中文总结 AI 辅助
给定1 < p < ∞，构造巴拿赫空间Y和闭凸对称集L⊆BY，使Y**强正则，L的w*闭包的非空相对弱星开子集半径为1，非空相对弱开子集直径至少为2^(1/p)，推进相关问题研究并给出部分答案。
中文摘要 AI 辅助
给定\(1 < p < \infty\)，我们构造一个巴拿赫空间\(Y\)和一个闭的、凸的且对称的集合\(L\subseteq B_Y\)，具有以下性质：1）\(Y^{**}\)是强正则的（因此\(Y\)是强正则的）。2）\(\overline{L}^{w^*}\)（\(L\)在\(Y^{**}\)中的\(w^*\)闭包）的每个非空相对弱星开子集半径为\(1\)。特别地，\(L\)的每个非空相对弱开子集半径为\(1\)。3）\(L\)的每个非空相对弱开子集直径至少为\(2^{\frac{1}{p}}\)。这推进了关于是否存在一个强正则巴拿赫空间，使得单位球的每个非空相对弱开子集半径为\(1\)的问题。作为部分答案，我们得到对于每个\(\varepsilon > 0\)，存在一个强正则巴拿赫空间，其中弱开子集半径至少为\(1 - \varepsilon\)。
英文摘要
We construct, given $1<p<\infty$, a Banach space $Y$ and a closed, convex and symmetric set $L\subseteq B_Y$ with the following properties: 1) $Y^{**}$ is strongly regular (henceforth, $Y$ is strongly regular). 2) Every non-empty relatively weakly-star open subset of $\overline{L}^{w^*}$ (the $w^*$ closure of $L$ in $Y^{**}$) has radius one. In particular, every non-empty relatively weakly open subset of $L$ has radius $1$. 3) Every non-empty relatively weakly open subset of $L$ has diameter, at least, $2^\frac{1}{p}$. This constitutes an advance to the question whether there exists a strongly regular Banach spaces satisfying that every non-empty relatively weakly open subset of the unit ball has radius $1$. As a partial answer, we get that for every $\varepsilon>0$ there exists a strongly regular Banach spaces where weakly open subsets have radius, at least, $1-\varepsilon$.
