ABSTRACT
This chapter discusses some properties of Hamel bases of the real line R and highlight their remarkable role in various constructions of strange additive functions acting from R into itself. The construction of a Hamel basis can be done by starting with one general theorem of the theory of vector spaces (over arbitrary fields). According to this general theorem, for every vector space E, there exists a basis of E, i.e., a maximal linearly independent subset of E. This assertion follows almost immediately from the Zorn lemma or, equivalently, from the Axiom of Choice. The chapter presents a very simple argument based on the Steinhaus property for the Lebesgue measure to prove that all nontrivial solutions of Cauchy’s functional equation are Lebesgue nonmeasurable and do not possess the Baire property. It also discusses the weakest form of so-called Jensen’s inequality, which plays a basic role in the theory of real-valued convex functions and is closely connected with Cauchy’s functional equation.
