On functional equations related to additive mappings and isometries (Q2340859)

The author studies certain functional equations related to the problem of characterization of metrics generated by norms. Among several interesting results, he proves the following theorem: Let \((X, +)\) be an abelian \(2\)-divisible group and \(Y\) be a real normed space. If \(f : X \to Y\) satisfies \(\|f(\frac{x-y}{2})\|=\frac{1}{2} \|f(x)- f(y)\|\) for all \(x,y\in X\), then there exists a nonempty set \(T \subset \mathbb{R}^X\), an additive operator \(A: X \to B(T,\mathbb{R})\) and an isometry \(I : A(X) \to Y\) with \(I(0) = 0\) such that \(f = I \circ A\). Recall that \(\mathbb{R}^X\) is the set of all real-valued mappings defined on \(X\) and \(B(T,\mathbb{R})\) is the linear space of all bounded mappings from a set \(T\) to \(\mathbb{R}\) equipped with the supremum norm.