We recall that a Hamel basis is any base of the linear space (ℝN; 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 AC ⊂ ℝN such that A is linearly independent over ℚ, and E(C) = ℝN, there exists a Hamel basis H of ℝN such that AHC. In particular, every set belonging to any of the classes A = B, ℭ, D(D), AC, BC 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 classesA = B, C, D(D), AC, BC.


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