Properties of Hamel Bases

Abstract

We recall that a Hamel basis is any base of the linear space (ℝ^N; ℚ; +; ·). We have constructed Hamel bases already many times in this book. Theorem 4.2.1 (cf., in particular, Corollary 4.2.1) asserts that there exist Hamel bases. More exactly (Lemma 4.2.1), for every set A ⊂ C ⊂ ℝ^N such that A is linearly independent over ℚ, and E(C) = ℝ^N, there exists a Hamel basis H of ℝ^N such that A ⊂ H ⊂ C. In particular, every set belonging to any of the classes A = B, ℭ, D(D), A _C , B _C contains a Hamel basis (Theorems 9.3.6 and 10.7.3 and Exercise 10.7). On the other hand, we have the following Theorem 11.1.1. No Hamel basis belongs to any of the classes A = B, C, D(D), A _C , B _C .