Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

Abstract

Some versions of the fundamental principles of the functional analysis in asymmetric normed spaces – the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Principle – are proved. The proofs are based on an asymmetric version of a lemma of Zabreijko (1969) [12] on the continuity of the countably subadditive functionals. At the same time a flaw in the proof of the Uniform Boundedness Principle, given in the book by Cobzaş (2013) [5], is fixed.