On some characterization of the absolute value of an additive function

Józef Tabor

Published: 1994-09-30

Vol. 8 (1994)

On some characterization of the absolute value of an additive function

Józef Tabor

Abstract

Let G be an abelian group, let K be the real or complex field, let X be a normed space over ????, and let u∈X such that ‖u‖=1 be given. We assume that there exists a subspace X₁ of X such that
X = Lin(u) ⊕ X₁
and
‖αu+x₁‖ ≥ max(|α|, ‖x₁‖) for α∈K , x₁∈X₁.
Then we prove that the general solution f: G→K of the equation
f(x+y) + f(x-y) + ‖f(x+y)-f(x-y)‖u = 2f(x) + 2f(y) for x,y∈G
is given by the formula
f(x) = |a(x)|u for x∈G,
where a: G→ℝ is an additive function.

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Tabor, J. (1994). On some characterization of the absolute value of an additive function. Annales Mathematicae Silesianae, 8, 69–77. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14218

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Vol. 8 (1994)
Published: 1994-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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