On some normability conditions

Abstract

Various normability conditions of locally convex spaces (including Vogt interpolation classes DNϕ and Ωϕ as well as quasi-and asymptotic normability) are investigated. In particular, it is shown that on the class of Schwartz spaces the property of asymptotic normability coincides with the property GS, which is a natural generalization of Gelfand-Shilov countable normability (cf. [9,25], where the metrizable case was treated). It is observed also that there are certain natural duality relationships among some of normability conditions.