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 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.
References
2. W. Mlak, Introduction to the theory of Hilbert spaces, (in Polish), PWN, Warszawa 1987.
Wyższa Szkoła Pedagogiczna im. Komisji Edukacji Narodowej w Krakowie Poland
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