On extension of (pseudo-)metrics
 from a subgroup
 of a topological group onto the group

O.V. Ravsky

In this paper we obtain necessary and in some cases sufficient conditions for existence of an extension of a two-side invariant metric from a~subgroup of a SIN-group onto the group. In particular, it is found a simple criterion for existence of extension of a bounded metric from a normal subgroup onto the group. It is also proved that every metric from an open subgroup of an abelian group can be extended onto the group.

1. Гуран I.Й., Зарiчний М.М. Елементи теорiї топологiчних груп. – Київ: НМК ВО, 1991.– 76 c.

2. Энгелькинг Р., Общая топология – М.,: Мир, 1981.

3. Bessaga C. Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam, 1992.– P.247-257.

4. Hausdorff F. Erweiterung einer Homömorpie Fund. Math. – 1930.– V.16.– P.353–360.

5. Isbell J.R. On finite-dimensional uniform spaces Pacific J. of Math.– 1959.– V.9.– P.107–121.

6. Roelcke W., Dierolf S. Uniform structures on topological groups and their quotiens, Mc Graw Hill, 1981.– 276 p.

7. Zarichnyi M. Regular linear operators extending metrics: a short proof Bull. Pol. Ac.:Math.– 1996.– V.44.– P.267–269.

	On extension of (pseudo-)metrics from a subgroup of a topological group onto the group
Author	O.V. Ravsky Department of Mechanics and Mathematics,Lviv State UniversityUniversytetska str. 1,290602, Lviv, Ukraine
Abstract	In this paper we obtain necessary and in some cases sufficient conditions for existence of an extension of a two-side invariant metric from a~subgroup of a SIN-group onto the group. In particular, it is found a simple criterion for existence of extension of a bounded metric from a normal subgroup onto the group. It is also proved that every metric from an open subgroup of an abelian group can be extended onto the group.
Keywords	two-sided invariant metric, SIN-group, metric extension, subgroup, normal subgroup, bounded metric
DOI	doi:10.30970/ms.11.1.31-40
Reference	1. Гуран I.Й., Зарiчний М.М. Елементи теорiї топологiчних груп. – Київ: НМК ВО, 1991.– 76 c. 2. Энгелькинг Р., Общая топология – М.,: Мир, 1981. 3. Bessaga C. Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam, 1992.– P.247-257. 4. Hausdorff F. Erweiterung einer Homömorpie Fund. Math. – 1930.– V.16.– P.353–360. 5. Isbell J.R. On finite-dimensional uniform spaces Pacific J. of Math.– 1959.– V.9.– P.107–121. 6. Roelcke W., Dierolf S. Uniform structures on topological groups and their quotiens, Mc Graw Hill, 1981.– 276 p. 7. Zarichnyi M. Regular linear operators extending metrics: a short proof Bull. Pol. Ac.:Math.– 1996.– V.44.– P.267–269.
Pages	31-40
Volume	11
Issue	1
Year	1999
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On extension of (pseudo-)metrics from a subgroup of a topological group onto the group