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function decomposition

Source: miklos schweitzer 1997 q7

September 24, 2021
group theoryEuclidean spacefunction

Problem Statement

Let G be an abelian group, 0≤ε<10\leq\varepsilon<1 and f:G→Rnf : G\to\Bbb R^n a function that satisfies the inequality. ∣∣f(x+y)−f(x)−f(y)∣∣≤ε∣∣f(y)∣∣(x,y)∈G2||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2 Prove that there is an additive function A:G→RnA : G\to \Bbb R^n and a continuous function φ:A(G)→Rn\varphi : A (G) \to\Bbb R^n such that f=φ∘Af = \varphi\circ A.
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