publication . Preprint . 2016

Free topological vector spaces

Gabriyelyan, Saak S.; Morris, Sidney A.;
Open Access English
  • Published: 13 Apr 2016
Abstract
We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_\omega$-space if and only if $X$ is a $k_\omega$-space. If $X$ is infinite, then $\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\mathbb{V}(\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable...
Subjects
free text keywords: Mathematics - General Topology, 46A03, 54A25, 54D50
Download from

1. A. V. Arhangel'skii, Topological function spaces, Math. Appl. 78, Kluwer Academic Publishers, Dordrecht, 1992.

2. A. V. Arhangel'skii, O. G. Okunev, V. G. Pestov, Free topological groups over metrizable spaces, Topology Appl. 33 (1989), 63-76. [OpenAIRE]

3. A. V. Arhangel'skii, M. G. Tkachenko, Topological groups and related strutures, Atlantis Press/World Scientific, Amsterdam-Raris, 2008.

4. T. Banakh, Fans and their applications in General Topology, Functional Analysis and Topological Algebra, available in arXiv:1602.04857.

5. W. Banaszczyk, Additive subgroups of topological vector spaces, LNM 1466, Berlin-Heidelberg-New York 1991.

6. V. K. Bel'nov, On dimension of free topological groups, Proc. IVth Tiraspol Symposium on General Topology, (1979), 14-15 (in Russian).

7. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

8. M. Fabian, P. Habala, P. H´ajek, V. Montesinos, J. Pelant, V. Zizler, Banach space theory. The basis for linear and nonlinear analysis, Springer, New York, 2010.

9. J. Flood, Free topological vector spaces, Ph. D. thesis, Australian National University, Canberra, 109 pp., 1975.

10. J. Flood, Free locally convex spaces, Dissertationes Math CCXXI, PWN, Warczawa, 1984.

11. S. Gabriyelyan, The k-space property for free locally convex spaces, Canadian Math. Bull. 57 (2014), 803-809.

12. S. Gabriyelyan, A characterization of free locally convex spaces over metrizable spaces which have countable tightness, Scientiae Mathematicae Japonicae 78 (2015), 201-205.

Abstract
We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_\omega$-space if and only if $X$ is a $k_\omega$-space. If $X$ is infinite, then $\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\mathbb{V}(\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable...
Subjects
free text keywords: Mathematics - General Topology, 46A03, 54A25, 54D50
Download from

1. A. V. Arhangel'skii, Topological function spaces, Math. Appl. 78, Kluwer Academic Publishers, Dordrecht, 1992.

2. A. V. Arhangel'skii, O. G. Okunev, V. G. Pestov, Free topological groups over metrizable spaces, Topology Appl. 33 (1989), 63-76. [OpenAIRE]

3. A. V. Arhangel'skii, M. G. Tkachenko, Topological groups and related strutures, Atlantis Press/World Scientific, Amsterdam-Raris, 2008.

4. T. Banakh, Fans and their applications in General Topology, Functional Analysis and Topological Algebra, available in arXiv:1602.04857.

5. W. Banaszczyk, Additive subgroups of topological vector spaces, LNM 1466, Berlin-Heidelberg-New York 1991.

6. V. K. Bel'nov, On dimension of free topological groups, Proc. IVth Tiraspol Symposium on General Topology, (1979), 14-15 (in Russian).

7. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

8. M. Fabian, P. Habala, P. H´ajek, V. Montesinos, J. Pelant, V. Zizler, Banach space theory. The basis for linear and nonlinear analysis, Springer, New York, 2010.

9. J. Flood, Free topological vector spaces, Ph. D. thesis, Australian National University, Canberra, 109 pp., 1975.

10. J. Flood, Free locally convex spaces, Dissertationes Math CCXXI, PWN, Warczawa, 1984.

11. S. Gabriyelyan, The k-space property for free locally convex spaces, Canadian Math. Bull. 57 (2014), 803-809.

12. S. Gabriyelyan, A characterization of free locally convex spaces over metrizable spaces which have countable tightness, Scientiae Mathematicae Japonicae 78 (2015), 201-205.

Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue