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Approximate Riesz Algebra‐Valued Derivations

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  • 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) xn → x and xn · y → z in ||·||v, then z = x · y; (b) yn → y and x · yn → 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.

Suggested Citation

  • Faruk Polat, 2012. "Approximate Riesz Algebra‐Valued Derivations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:240258
    DOI: 10.1155/2012/240258
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    References listed on IDEAS

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    1. Faruk Polat, 2012. "Some Generalizations of Ulam‐Hyers Stability Functional Equations to Riesz Algebras," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Faruk Polat, 2012. "Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-9, January.
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