AI 中文总结
该研究针对$C_p(X)$空间的商空间相关公开问题，证明了若$C_p(X)$到带逐点拓扑的$c_0$对称序列理想有连续线性满射，则该理想必为$c_0$本身，结合已有结果给出了满射存在的完整刻画。
AI 中文摘要
Rosenthal经典定理指出，对任意无限紧空间$X$，Banach空间$C(X)$都有同构于$c_0$或$\u2113_2$的商空间。而对于赋予逐点收敛拓扑的空间$C_p(X)$，对应的问题要微妙得多，目前仍未解决；在ZFC公理系统中，目前尚未确定是否存在无限维可度量化商空间的紧空间仅有Efimov紧集。我们证明，Banach空间的情形无法在带逐点拓扑的常用序列空间中复现：设$X$为Tychonoff空间，$E\subseteq c_0$为赋予$\mathbb{R}^{\mathbb{N}}$子空间拓扑的非平凡对称序列理想，则存在连续线性满射$T:C_p(X)\rightarrow E_p$蕴含$E=c_0$，其中$E_p$表示赋予$\mathbb{R}^{\mathbb{N}}$子空间拓扑的$E$。由此可知，$c_0$的任何非零真对称序列理想都不能实现为$C_p$空间的连续线性像。将该结果与Banakh、Kąkol和Śliwa得到的$C_p(X)$的Josefson--Nissenzweig性质的刻画相结合，我们推导出了所有存在此类满射的配对$(X,E)$的完整刻画。特别地，对任意$0<q<\infty$，不存在连续线性满射$C_p(X)\rightarrow(\u2113_q)_p$。
英文摘要
Rosenthal's classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or $\ell_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from $\mathbb{R}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\rightarrow E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from $\mathbb{R}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson--Nissenzweig property for $C_p(X)$ obtained by Banakh, Kąkol, and Śliwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\infty$, there is no continuous linear surjection $C_p(X)\rightarrow(\ell_q)_p$.
