Eine zur parallelogrammgleichung äquivalente ungleichung
A Gilányi - Aequationes Mathematicae, 2001 - Springer
Aequationes Mathematicae, 2001•Springer
In this paper it is proved that, for a function\(f: G\to E\) mapping from an abelian group G
divisible by 2 into an inner product space E, the functional inequality¶¶\(\Vert2f (x)+ 2f (y)-f
(xy^{-1})\Vert\leq\Vert f (xy)\Vert\\\(x, y\in G)\)¶ implies the parallelogram equation¶\(f (xy)+ f
(xy^{-1})-2f (x)-2f (y)= 0\\\(x, y\in G)\).
divisible by 2 into an inner product space E, the functional inequality¶¶\(\Vert2f (x)+ 2f (y)-f
(xy^{-1})\Vert\leq\Vert f (xy)\Vert\\\(x, y\in G)\)¶ implies the parallelogram equation¶\(f (xy)+ f
(xy^{-1})-2f (x)-2f (y)= 0\\\(x, y\in G)\).
Summary
In this paper it is proved that, for a function mapping from an abelian group G divisible by 2 into an inner product space E, the functional inequality¶¶\(\Vert2f (x)+ 2f (y)-f (xy^{-1})\Vert\leq\Vert f (xy)\Vert\\\(x, y\in G)\)¶ implies the parallelogram equation¶\(f (xy)+ f (xy^{-1})-2f (x)-2f (y)= 0\\\(x, y\in G)\).
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