On some conditional functional equations
Abstract
Let X, Y be real linear spaces. We are looking for a function G : X2→ℝ such that the equation
f(x+y) = G(x,y)[f(x)+f(y)]
is equivalent to the orthogonal Cauchy equation
x⟂y ⇒ f(x+y)=f(x)+f(y).
Several kinds of orthogonalities are considered. The quotient ‖(x-y)/(x+y)‖ closely connected with the James orthogonality plays here a distinguished role. Similar problems are considered for the Ptolemaic equation
x⟂y ⇒ f(x+y)f(x-y) = f(x)2+f(y)2.
As a result a characterization of inner product spaces is obtained.
Keywords
conditional functional equations; orthogonal equations; James orthogonality; Birkhoff-James orthogonality
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Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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