## Abstract

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), 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) → C_{p}(Y) and C_{p}(X) is distinguished, then C_{p}(Y) also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator T: C_{p}(X) → 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]) → 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 Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of Δ -spaces we prove that a linear continuous surjection T: C_{p}(X) → 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⊂ R^{n}).

Original language | English |
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Article number | 199 |

Journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas |

Volume | 115 |

Issue number | 4 |

DOIs | |

State | Published - 1 Oct 2021 |

## Keywords

- Countable ordinal
- Distinguished locally convex space
- Linear continuous operator
- Δ -space

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Computational Mathematics
- Applied Mathematics

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