On some characterization of the absolute value of an additive function



Abstract

Let G be an abelian group, let 𝕂 be the real or complex field, let X be a normed space over 𝕂, and let uX 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, x1X1.
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,yG
is given by the formula
f(x) = |a(x)|u    for xG,
where a: G→ℝ is an additive function.


1. A. Chaljub-Simon, P. VoIkmann, Caractérisation du module d'une fonction additive à l'aide d'une d'équation fonctionnelle, Aequationes Math. 47 (1994) 60-68.
2. W. Mlak, Introduction to the theory of Hilbert spaces, (in Polish), PWN, Warszawa 1987.
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Published : 1994-09-30


TaborJ. (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

Józef Tabor 
Wyższa Szkoła Pedagogiczna im. Komisji Edukacji Narodowej w Krakowie  Poland



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