Non - separable Banach spaces with non - meager Hamel basis

We show that an infinite-dimensional complete linear space X has: • a dense hereditarily Baire Hamel basis if |X| ≤ c; • a dense non-meager Hamel basis if |X| = κ = 2 for some cardinal κ. According to Corollary 3.4 of [BDHMP] each infinite-dimensional separable Banach space X has a non-meager Hamel basis. This is a special case of Theorem3.3 of [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 abovementioned 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 open dense subsets of F is dense in F ). Our next result treats Banach spaces of even larger size. We define a subset A of a topological space X to be κ-perfect for some cardinal κ if each non-empty open set U of A has size |U | ≥ κ. Note that a Hausdorff space X is ω-perfect if and only if it has no isolated points (so is perfect in the standard sense). It is well-known (see [BDHMP, 2.8]) that each Banach space X has size |X| = d(X)ω. Our second principal result generalizes Theorem 3.3 of [BDHMP]. 2000 Mathematics Subject Classification: 46B15, 03E75.