On vector spaces and algebras with maximal locally pseudoconvex topologies

A. Kokk; W. Żelazko

Studia Mathematica (1995)

  • Volume: 112, Issue: 2, page 195-201
  • ISSN: 0039-3223

Abstract

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Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras, considering the problem of uniqueness of a complete topology for semitopological algebras and giving an example of a complete locally convex commutative semitopological algebra without multiplicative linear functionals, but with every separable subalgebra having a total family of such functionals.

How to cite

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Kokk, A., and Żelazko, W.. "On vector spaces and algebras with maximal locally pseudoconvex topologies." Studia Mathematica 112.2 (1995): 195-201. <http://eudml.org/doc/216146>.

@article{Kokk1995,
abstract = {Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras, considering the problem of uniqueness of a complete topology for semitopological algebras and giving an example of a complete locally convex commutative semitopological algebra without multiplicative linear functionals, but with every separable subalgebra having a total family of such functionals.},
author = {Kokk, A., Żelazko, W.},
journal = {Studia Mathematica},
keywords = {maximal -convex topology; complete Hausdorff topological vector space; complete locally pseudoconvex space; topological algebras; complete locally convex commutative semitopological algebra without multiplicative linear functionals},
language = {eng},
number = {2},
pages = {195-201},
title = {On vector spaces and algebras with maximal locally pseudoconvex topologies},
url = {http://eudml.org/doc/216146},
volume = {112},
year = {1995},
}

TY - JOUR
AU - Kokk, A.
AU - Żelazko, W.
TI - On vector spaces and algebras with maximal locally pseudoconvex topologies
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 2
SP - 195
EP - 201
AB - Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras, considering the problem of uniqueness of a complete topology for semitopological algebras and giving an example of a complete locally convex commutative semitopological algebra without multiplicative linear functionals, but with every separable subalgebra having a total family of such functionals.
LA - eng
KW - maximal -convex topology; complete Hausdorff topological vector space; complete locally pseudoconvex space; topological algebras; complete locally convex commutative semitopological algebra without multiplicative linear functionals
UR - http://eudml.org/doc/216146
ER -

References

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  1. [1] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. Zbl0437.26007
  2. [2] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, 1972. 
  3. [3] H. Schaefer, Topological Vector Spaces, Springer, New York, 1971. 
  4. [4] L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Math. 230, Springer, 1971. 
  5. [5] W. Żelazko, On certain open problems in topological algebras, Rend. Sem. Mat. Fis. Milano 59 (1989), 1992, 49-58. Zbl0755.46019
  6. [6] W. Żelazko, On topologization of countably generated algebras, Studia Math. 112 (1994), 83-88. Zbl0832.46042
  7. [7] W. Żelazko, Further examples of locally convex algebras, in: Topological Vector Spaces, Algebras and Related Areas, Pitman Res. Notes in Math., to appear. Zbl0887.46028

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