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On approximation by values of additive functions. (Über Approximation durch Werte additiver Funktionen.) (German) Zbl 0785.39003

The author offers novel proofs of the classical results that additive functions on reals, bounded from one side on an interval are linear, while the graph of a nonlinear additive function is everywhere dense in the real plane, and also similar results on vector spaces.
Fundamental is the following observation. Let \(f\) be (throughout this review) an additive function from the vector space \(E\) into the vector space \(F\). If \(E\) and \(F\) are real vector spaces, \(r\) a real number and \(x\) an element of \(E\) then \(f(rx)-rf(x) = f(\{r\}x)-\{r\}f(x)\), where \(\{r\}\) is the fractional part of \(r\). We denote by \(D(f;r)(x)\) the left hand side of this equality and by \(D(f;F)\) the subspace of \(F\) generated by \(D(f;r)(x)\). If \(E\) and \(F\) are topological vector spaces and the image of \(E\) under \(f\) is a subset of \(D(f;F)\) then the graph of the nonlinear \(f\) is dense in the cartesian product of \(E\) and of \(D(f;F)\). If \(f\) is continuous at a point of the topological vector space \(E\) then it is linear. If \(E\) and \(F\) are normed spaces and \(f\) bounded in a neighborhood of one point then it is linear and continuous.
Several examples are offered of “small” sets \(S\) of real numbers such that the graph of every nonlinear real additive function on \(S\) is dense on the cartesian product of \(S\) and of the set of real numbers.
{Note: The remarkable proof by G. S. Young, quoted on p. 164, concerned additive functions bounded on a proper interval from both sides. However, the generalization to functions bounded from one side is easy}.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
46B03 Isomorphic theory (including renorming) of Banach spaces
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