NON-SEPARABLE BANACH SPACES WITH NON-MEAGER HAMEL BASIS

Abstract. We show that an infinite-dimensional complete linear spaceX has: • a dense hereditarily Baire Hamel basis if |X | ≤ c+; • a dense non-meager Hamel basis if |X | = κω = 2κ for some cardinal κ. According to Corollary 3.4 of [BDHMP] each infinite-dimensional separable Ba-nach space X posesses a non-meager Hamel basis. This is a special case of The-orem 3.3 [BDHMP] asserting that an infinite-dimensional Banach space X has a non-meager Hamel basis provided 2d(X) = d(X)ω, where d(X) is the density of X. Having in mind those results the authors of [BDHMP] asked if each infinite-dimensional Banach space has a non-meager Hamel basis. In this paper we shall give two partial answers to this question generalizing the mentioned Corollary 3.4 and Theorem 3.3 of [BDHMP] in two directions. Theorem 1. Each infinite-dimensional linear complete metric space X of size |X | ≤ c+ has a dense hereditarily Baire Hamel basis. We recall that a topological space X is hereditarily Baire if each closed subspace F of X is Baire (in the sense that the intersection of a countable family of ope