## Abstract

The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x∈ X there is a base { U_{α}: α∈ N^{N}} of neighborhoods at x such that U_{β}⊆ U_{α} whenever α≤ β for all α, β∈ N^{N}, where α= (α(n)) _{n} _{∈} _{N}≤ β= (β(n)) _{n} _{∈} _{N} if α(n) ≤ β(n) for all n∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of N^{N}-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.

Original language | English |
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Pages (from-to) | 575-585 |

Number of pages | 11 |

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

Volume | 111 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2017 |

## Keywords

- C(X)
- Compact resolution
- Free locally convex space
- G-base

## ASJC Scopus subject areas

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