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.
@article{bwmeta1.element.bwnjournal-article-smv112i2p195bwm,
author = {A. Kokk and W. \.Zelazko},
title = {On vector spaces and algebras with maximal locally pseudoconvex topologies},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {195-201},
zbl = {0837.46037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv112i2p195bwm}
}
Kokk, A.; Żelazko, W. On vector spaces and algebras with maximal locally pseudoconvex topologies. Studia Mathematica, Tome 113 (1995) pp. 195-201. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv112i2p195bwm/
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