Free topological vector spaces

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.