Ph.D. Qualifying Exam – Spring 2004

2 Let X be an infinite dimensional Banach space and suppose it has a countable Hamelbasis (x n ) n∈N ⋆. Letso that∀n ∈ N ⋆ X n = Span (x 1 , . . .,x n )X = ⋃n∈N ⋆ X nEach X n is closed. Indeed, let (u p ) p∈N be a sequence in X n , converging to some u ∈ X.Then (u p ) p∈N is Cauchy; but we know that finite dimensional vector spaces are complete,thus (u p ) p∈N has a limit v ∈ X n . By unicity of the limit, u = v and therefore u ∈ X n .By Baire’s lemma, there has to be a positive integer N such that X N has nonemptyinterior: there exists x ∈ X N and ǫ > 0, such that B(x, 2ǫ) ⊂ X N . We should deduce fromthis that X N = X. Let y ∈ X be nonzero. Then x + ǫ y is in the ball B(x, 2ǫ) ⊂ X ‖y‖ N.Therefore, since X N is a vector space,[ ‖y‖y =ǫwhich proves(x + ǫ‖y‖ y )− ‖y‖ǫ x ]∈ X NX ⊂ X NSo X is finite dimensional and we have a contradiction.A Hamel basis for an infinite dimensional Banach space is uncountable.16