# On vector spaces and algebras with maximal locally pseudoconvex topologies

Studia Mathematica (1995)

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

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topKokk, 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

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

## Citations in EuDML Documents

top- W. Żelazko, A non-locally convex topological algebra with all commutative subalgebras locally convex
- W. Żelazko, A non-Banach in-convex algebra all of whose closed commutative subalgebras are Banach algebras.
- W. Żelazko, The strongest vector space topology is locally convex on separable linear subspaces
- M. Wojciechowski, W. Żelazko, Non-uniqueness of topology for algebras of polynomials

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