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)
On approximate derivations. (English) Zbl 1093.39024
The author of the present pleasant paper establishes that if $A$ is a subalgebra of a Banach algebra $B$ and $f: A \to B$ satisfies $\Vert f(x+y) - f(x) - f(y)\Vert \leq \delta$ and $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon$, for all $x, y \in A$ and for some $\delta, \varepsilon \geq 0$, then there exists a unique additive derivation $d:A \to B$ such that $\Vert f(x) - d(x)\Vert \leq \delta \quad (x \in A)$, and $x\left(f(y) - d(y)\right) = 0 \quad (x, y \in A)$. The result and its proof are still true for a more general case if we consider a normed algebra $A$ and replace $B$ by a Banach $A$-bimodule $X$. He also proves that if $B$ is a normed algebra with an identity belonging to $A$, then every mapping $f: A \to B$ satisfying $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon \quad (x, y \in A)$ must fulfil $f(xy) = xf(y) - f(x)y \quad (x, y \in A)$. This superstability result is nice, since there is not assumed any (approximately) additive condition on $f$. Some similar results in which one considers generalized derivations can be found in the reviewer’s paper [“Hyers-Ulam-Rassias stability of generalized derivations”, Int. J. Math. Math. Sci. (to appear)].

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
47B47Commutators, derivations, elementary operators, etc.