关于区别空间$C_p(X)$之间的线性连续算子
On linear continuous operators between distinguished spaces $C_p(X)$
摘要：
如[16]中所证明的，对于吉洪诺夫空间$X$，局部凸空间$C_{p}(X)$是可区分的当且仅当$X$是$Δ$ -空间。若存在线性连续满射$T:C_p(X) \to C_p(Y)$，且$C_p(X)$可判别，则$C_p(Y)$也可判别[17].本文首先讨论了以下问题：在什么条件下上述算子$T:C_p(X) \to C_p(Y)$是开的？其次，我们特别关注具体的区别空间$C_p([1,α])$，其中$α$是可数序数。给出了所有容许线性连续满射映射$T:C_p([1,α]) \to C_p(Y)$的$Y$的一个完全刻划.我们还观察到，对于每一个可数序数$α$，$C_p([1,α])$的所有闭线性子空间都是可区分的，从而回答了[17]中提出的一个公开问题。利用$Δ$ -空间的一些性质，证明了对于每个无限可度量化紧$C$-空间$X$（特别是对于每个无限紧$X \subset \mathbb{R}^n$），不存在线性连续满射$T:C_p(X) \to C_k(X)_w$，其中$C_k(X)_w$表示赋有弱拓扑的Banach空间$C(X)$.
英文摘要：
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Δ$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) \to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,α])$, where $α$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,α]) \to C_p(Y)$ is given. We also observe that for every countable ordinal $α$ all closed linear subspaces of $C_p([1,α])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Δ$-spaces we prove that a linear continuous surjection $T:C_p(X) \to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X \subset \mathbb{R}^n$).
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