- Title
- Free topological vector spaces
- Creator
- Gabriyelyan, Saak; Morris, Sidney
- Date
- 2017
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/156253
- Identifier
- vital:11389
- Identifier
-
https://doi.org/10.1016/j.topol.2017.03.006
- Identifier
- ISSN:0166-8641
- 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 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.
- Publisher
- Elsevier B.V.
- Relation
- Topology and its Applications Vol. 223, no. (2017), p. 30-49
- Rights
- Copyright © 2017 Elsevier B.V.
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; Barreled space; Free locally convex space; Free topological vector space; kω-Space
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