AI 中文总结
该研究证明紧豪斯多夫空间上的逐点函数空间$C_p(K)$与弱巴拿赫空间不同胚，解决了相关拓扑学问题并回应了特定疑问。
AI 中文摘要
设$K$为紧豪斯多夫空间，$E$为无限维实巴拿赫空间，我们证明不存在连续双射$h\colon C_p(K)\to E_w$，其逆映射在$h(0)$处连续。由此，对任意无限紧豪斯多夫空间$K$和$L$，$C_p(K)$与$C_w(L)$不同胚。这解决了Krupski问题及其Krupski与Marciszewski提出的两空间版本，并回答了Kąkol、Leiderman和Michalak关于$C_p([0,1])$与弱巴拿赫空间的问题。
英文摘要
Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.
Comments5 pp
