Embedding into free topological vector spaces on compact metrizable spaces
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2018
- Type: Text , Journal article
- Relation: Topology and its Applications Vol. 233, no. (2018), p. 33-43
- Full Text: false
- Reviewed:
- Description: For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by I, G, Q and Sk the closed unit interval, the Cantor space, the Hilbert cube Q=IN and the k-dimensional unit sphere for k
Free topological vector spaces
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2017
- Type: Text , Journal article
- Relation: Topology and its Applications Vol. 223, no. (2017), p. 30-49
- Full Text: false
- Reviewed:
- Description: 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 kω-space if and only if X is a kω-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. © 2017 Elsevier B.V.