Author’s abstract: “We study the stability of homomorphisms between topological (abelian) groups. Inspired by the “singular” case in the stability of Cauchy’s equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps such that and
(in ) as in . The basic question here is whether is approximable by a true homomorphism in the sense that in as in . Our main result is that quasi-homomorphisms are approximable in the following two cases:
is a product of locally compact abelian groups and is either or the circle group .
is either or and is a Banach space.
This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions.”
A reference for both main concepts and results in the subject is the book of D. H. Hyers, G. Isac and Th. M. Rassias [Stability of functional equations in several variables (Progress in Nonlinear Differential Equations and their Applications 34, Boston, Birkhäuser) (1998; Zbl 0907.39025)].