Some Counterexamples Concerning Strong M-Bases of Banach Spaces Original Research Article

A sequence of elements (ƒn)∞1 of a real or complex Banach space X is an M-basis (of X) if (V∞n = 1ƒn = X and there exists a biorthogonal sequence of elements (ƒ*n)∞1 of X* satisfying ∩∞n=1ker ƒ*n = (0). An M-basis (ƒn)∞1 is a strong M-basis if, additionally, x ∈ ∨{ƒn : ƒ*n(x) ≠ 0}, for every element x ∈ X. Let X be a Banach space having a (Schauder) basis. We show that there exists a strong M-basis of X which is not finitely series summable. It follows that there is an atomic Boolean subspace lattice on X, with one-dimensional atoms, that fails to have the strong rank one density property. We show that there is always an atomic Boolean subspace lattice on X, with precisely four atoms, that also fails to have this density property. Also, if X = c0 or c, an example is given of a strong M-basis (ƒn)∞1 of X such that V∞n=1ƒ*n = X* but with (ƒ*n)∞1 failing to be a strong M-basis of X*. This partially follows from a description that is given of a class of strong M-bases of c0, c, lp (1 ≤ p < ∞).