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 $Delta$-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 $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-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 $X subset mathbb{R}^n$).
تحميل البحث