Various normability conditions of locally convex spaces (including Vogt interpolation classes DN phi and Omega phi 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. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.