On vector spaces and algebras with maximal locally pseudoconvex topologies

Tom 112 / 1995

A. Kokk, W. Żelazko Studia Mathematica 112 (1995), 195-201 DOI: 10.4064/sm-112-2-195-201


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.


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