Abstract
We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}} \neq\emptyset}\) for every infinite set \({{\mathcal{H}} \subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.
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Dedicated to Professor Roman Ger on his 70th birthday.
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Jabłoński, W. Additive iterative roots of identity and Hamel bases. Aequat. Math. 90, 133–145 (2016). https://doi.org/10.1007/s00010-015-0376-9
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DOI: https://doi.org/10.1007/s00010-015-0376-9