zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quasi-homomorphisms. (English) Zbl 1051.39032
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 $\omega:{\cal G}\to{\cal H}$ such that $\omega(0)=0$ and $$ \omega(x+y)-\omega(x)-\omega(y)\to 0 $$ (in ${\cal H}$) as $x,y\to 0$ in ${\cal G}$. The basic question here is whether $\omega$ is approximable by a true homomorphism $a$ in the sense that $\omega(x)-a(x)\to 0$ in ${\cal H}$ as $x\to 0$ in ${\cal G}$. Our main result is that quasi-homomorphisms $\omega:{\cal G}\to{\cal H}$ are approximable in the following two cases: $\bullet$ ${\cal G}$ is a product of locally compact abelian groups and ${\cal H}$ is either ${\Bbb R}$ or the circle group ${\Bbb T}$. $\bullet$ ${\cal G}$ is either ${\Bbb R}$ or ${\Bbb T}$ and ${\cal H}$ 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 {\it D. H. Hyers, G. Isac} and {\it 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)].

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
Full Text: DOI Link