Free topological vector spaces

2016/04/14 by Gabriyelyan, Saak S., Morris, Sidney A.
#46A03 #54A25 #54D50 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1604.04005

Abstract

We define and study the free topological vector space \mathbbV(X) over a Tychonoff space X. We prove that \mathbbV(X) is a k_ω-space if and only if X is a k_ω-space. If X is infinite, then \mathbbV(X) contains a closed vector subspace which is topologically isomorphic to \mathbbV(ℕ). It is proved that if X is a k-space, then \mathbbV(X) is locally convex if and only if X is discrete and countable. If X is a metrizable space it is shown that: (1) \mathbbV(X) has countable tightness if and only if X is separable, and (2) \mathbbV(X) is a k-space if and only if X is locally compact and separable. It is proved that \mathbbV(X) is a barrelled 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 quasibarrelled if and only if L(X) is barrelled if and only if X is discrete, and (2) L(X) is a Baire space if and only if X is finite.

Related

• A description of the topology of free topological vector spaces
• The k-property and countable tightness of free topological vector spaces
• Locally convex properties of free locally convex spaces
• The topological properties of q-spaces in free topological groups
• \mathfrak G-bases in free (locally convex) topological vector spaces