## Abstract

We continue our initial study of C_{p}(X) spaces that are distinguished, equiv., are large subspaces of R^{X}, equiv., whose strong duals L_{β}(X) carry the strongest locally convex topology. Many are distinguished, many are not. All L_{β}(X) spaces are, as are all metrizable C_{p}(X) and C_{k}(X) spaces. To prove a space C_{p}(X) is not distinguished, we typically compare the character of L_{β}(X) with |X|. A certain covering for X we call a scant cover is used to find distinguished C_{p}(X) spaces. Two of the main results are: (i) C_{p}(X) is distinguished if and only if its bidual E coincides with R^{X}, and (ii) for a Corson compact space X, the space C_{p}(X) is distinguished if and only if X is scattered and Eberlein compact.

Original language | English |
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Article number | 27 |

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

Volume | 115 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2021 |

## Keywords

- Bidual space
- Distinguished space
- Eberlein compact space
- Fréchet space
- Fundamental family of bounded sets
- G-dense subspace
- Point-finite family
- strongly splittable space

## ASJC Scopus subject areas

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

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