Approximate Riesz Algebra‐Valued Derivations

Faruk Polat

Abstract and Applied Analysis, 2012, vol. 2012, issue 1

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.

Date: 2012
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