Abstract and Applied Analysis

Approximate Riesz Algebra-Valued Derivations

Faruk Polat

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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 x and x n · y z in | | · | | v , then z = x · y ; (b) y n y and x · y n z in | | · | | v , then z = x · y . Let ε and δ > be two nonnegative real numbers. Assume that a map f : F F satisfies | | f ( x + y ) - f ( x ) - f ( y ) | | u ε and | | f ( x · y ) - x · f ( y ) - f ( x ) · y | | v δ for all x , y F . In this paper, we prove that there exists a unique derivation d : F F such that | | f ( x ) - d ( x ) | | u ε , ( x F ). Moreover, x · ( f ( y ) - d ( y ) ) = 0 for all x , y 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


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