As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, locally convex $$C_{p}(X)$$ is distinguished if only X $$\Delta $$ -space. If there exists linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ $$C_p(X)$$ distinguished, then $$C_p(Y)$$ also (Ka̧kol Proc B, 2021). Firstly, this paper we explore the following question: Under which conditions ...