Semitopological Vector Spaces and Hyperseminorms

In this paper, we introduce and study semitopological vector spaces. The goal is to provide an efficient 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. Sufficient 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