Additive involutions and Hamel bases

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 XX. We prove in a simple way that some classes of discontinuous additive involutions are dense in the topological vector space \({{\mathcal{A}}_X}\) .