Abstract
Our aim is to give other proofs of some slight generalizations of results from Baron (Aequat Math, 2013). We describe larger classes of discontinuous additive involutions \({a:X\to X}\) on a topological vector space X such that \({a(H)\setminus H\neq\emptyset}\) holds for a sufficiently numerous set \({H\subset X}\) of vectors linearly independent over \({{\mathbb{Q}}}\) . We also consider the topological vector space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced by the Tychonoff topology of the space X X. We prove in a simple way that some classes of discontinuous additive involutions are dense in the topological vector space \({{\mathcal{A}}_X}\) .
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Baron, K.: On additive involutions and Hamel bases. Aequat. Math. doi:10.1007/s00010-012-0183-5 (2013)
Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. In: Gilà ànyi, A. (ed.) 2nd edn. Birkhauser Verlag, Basel (2009)
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Jabłoński, W. Additive involutions and Hamel bases. Aequat. Math. 89, 575–582 (2015). https://doi.org/10.1007/s00010-013-0241-7
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DOI: https://doi.org/10.1007/s00010-013-0241-7