Orthogonality and additivity modulo \(\mathbb{Z}\)
Suppose that \(E\) is a real topological vector space. If \(f: E\to \mathbb{R}\) satisfies (1) \(f(x+ y)- f(x)- f(y)\in \mathbb{Z}\) for all \(x,y\in E\) and if there exist a non-empty open set \(U\subset E\) and a \(\gamma\in (0, 1/2)\) such that \(f(U)\subset (- \gamma, \gamma)+ \mathbb{Z}\), then there exists a continuous linear functional \(g: E\to \mathbb{R}\) such that \(f(x)- g(x)\in \mathbb{Z}\) for every \(x\in E\). The authors prove an analogous result by assuming (1) for orthogonal \(x\), \(y\) from a real inner product space and they apply it for orthogonally exponential functions. For the earlier result see for example the first author and \textit{P. Volkmann} [Abh. Math. Semin. Univ. Hamb. 61, 189-195 (1991; Zbl 0753.39002)].
scientific article; zbMATH DE number 785160