Semitopological Vector Spaces and Hyperseminorms

Abstract

In this paper, we introduce and study semitopological vector spaces. The goal is to provide an ecient base
for developing the theory of extrafunction spaces in an abstract setting of algebraic systems and topological spaces.
Semitopological vector spaces are more general than conventional topological vector spaces, which proved to be very
useful for solving many problems in functional analysis. To study semitopological vector spaces, hypermetrics and
hyperpseudometrics are introduced and it is demonstrated that hyperseminorms, studied in previous works of the
author, induce hyperpseudometrics, while hypernorms induce hypermetrics. Sucient and necessary conditions for
a hyperpseudometric (hypermetric) to be induced by a hyperseminorm (hypernorm) are found. We also show that
semitopological vector spaces are closely related to systems of hyperseminorms. Then defining boundedness and
continuity relative to associated systems of hyperseminorms, we study relations between relative boundedness and
relative continuity for mappings of vector spaces with systems of hyperseminorms and systems of hypernorms.