## Abstract and Applied Analysis

### Approximate Riesz Algebra-Valued Derivations

Faruk Polat

#### Abstract

Let $F$ be a Riesz algebra with an extended norm $||·|{|}_{u}$ such that $(F,||·|{|}_{u})$ is complete. Also, let $||·|{|}_{v}$ be another extended norm in $F$ weaker than $||·|{|}_{u}$ such that whenever (a) ${x}_{n}\to x$ and ${x}_{n}·y\to z$ in $||·|{|}_{v}$, then $z=x·y$; (b) ${y}_{n}\to y$ and $x·{y}_{n}\to z$ in $||·|{|}_{v}$, then $z=x·y$. Let $\epsilon$ and $\delta >$ be two nonnegative real numbers. Assume that a map $f:F\to F$ satisfies $||f(x+y)-f(x)-f(y)|{|}_{u}\le \epsilon$ and $||f(x·y)-x·f(y)-f(x)·y|{|}_{v}\le \delta$ for all $x,y\in F$. In this paper, we prove that there exists a unique derivation $d:F\to F$ such that $||f(x)-d(x)|{|}_{u}\le \epsilon$, ($x\in F$). Moreover, $x·(f(y)-d(y))=0$ for all $x,y\in F$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 240258, 5 pages.

Dates
First available in Project Euclid: 7 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1399487771

Digital Object Identifier
doi:10.1155/2012/240258

Mathematical Reviews number (MathSciNet)
MR2975349

Zentralblatt MATH identifier
1264.46035

#### Citation

Polat, Faruk. Approximate Riesz Algebra-Valued Derivations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 240258, 5 pages. doi:10.1155/2012/240258. https://projecteuclid.org/euclid.aaa/1399487771