On additive involutions and Hamel bases

Karol Baron

doi:10.1007/s00010-012-0183-5

Abstract

We provide an example of a discontinuous involutory additive function $${a: \mathbb{R}\to \mathbb{R}}$$ such that $${a(H) \setminus H \ne \emptyset}$$ for every Hamel basis $${H \subset \mathbb{R}}$$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from $${\mathbb{R}}$$ to $${\mathbb{R}}$$ with the Tychonoff topology induced by $${\mathbb{R}^{\mathbb{R}}}$$ .

Highlights

By a Hamel basis of R we mean a basis of the vector space R over the field Q of rationals
Inspired by the foot-note on p. 325 of [2] we are interested in discontinuous additive functions a : R → R which are involutory, i.e., a ◦ a = idR, and such that a(H)\H = ∅
The following theorem provides an example of a discontinuous involutory additive function a : R → R such that (1) holds for every set H ⊂ R which is linearly independent over Q and has at least three elements

Summary

Introduction

By a Hamel basis of R we mean (see [2, p. 82]) a basis of the vector space R over the field Q of rationals. 82]) a basis of the vector space R over the field Q of rationals. 325 of [2] we are interested in discontinuous additive functions a : R → R which are involutory, i.e., a ◦ a = idR, and such that a(H)\H = ∅.

Existence

Density