## Abstract

In this paper the free topological vector space V(X) over a Tychonoff space X is defined and studied. It is proved that V(X) is a k_{ω}-space if and only if X is a k_{ω}-space. If X is infinite, then V(X) contains a closed vector subspace which is topologically isomorphic to V(N). It is proved that for X a k-space, the free topological vector space V(X) is locally convex if and only if X is discrete and countable. The free topological vector space V(X) is shown to be metrizable if and only if X is finite if and only if V(X) is locally compact. Further, V(X) is a cosmic space if and only if X is a cosmic space if and only if the free locally convex space L(X) on X is a cosmic space. If a sequential (for example, metrizable) space Y is such that the free locally convex space L(Y) embeds as a subspace of V(X), then Y is a discrete space. It is proved that V(X) is a barreled topological vector space if and only if X is discrete. This result is applied to free locally convex spaces L(X) over a Tychonoff space X by showing that: (1) L(X) is quasibarreled if and only if L(X) is barreled if and only if X is discrete, and (2) L(X) is a Baire space if and only if X is finite.

Original language | English |
---|---|

Pages (from-to) | 30-49 |

Number of pages | 20 |

Journal | Topology and its Applications |

Volume | 223 |

DOIs | |

State | Published - 1 Jun 2017 |

## Keywords

- Barreled space
- Free locally convex space
- Free topological vector space
- k-Space

## ASJC Scopus subject areas

- Geometry and Topology