On Biorthogonal Systems and Mazur’s Intersection Property

We give a characterization of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ , fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur’s intersection property. A biorthogonal system in a Banach space X is a subset {xγ , fγ}γ∈Γ ⊂ X×X such that fγ(xγ′) = δγγ′ for γ, γ ′ ∈ Γ. The biorthogonal system {xγ, fγ}γ∈Γ in X is called fundamental if X = span{xγ; γ ∈ Γ}. A Markushevich basis is a fundamental biorthogonal system {xγ , fγ}γ∈Γ in X such that {fγ}γ∈Γ separates points of X. A Markushevich basis {xγ, fγ}γ∈Γ ⊂ X ×X ∗ is called shrinking if X = span{fγ; γ ∈ Γ}. In this note we use Γ as a cardinal number. A Banach space X is said to be an Asplund space, if every separable subspace of X has a separable dual. A Banach space X has Mazur’s intersection property if every bounded closed convex set can be represented as an intersection of closed balls. A density of a topological space is the least cardinality of a dense set. We refer to [2] for undefined terms used in this paper. It is known, [9, Theorem 7.18, Theorem 7.12], that if a dual unit ball of a Banach space X is a Corson compact, then densX = w-dens X and the following are equivalent. (i) X has a shrinking Markushevich basis, (ii) X is an Asplund space, (iii) X admits a Fréchet smooth norm. Let us remark that if a norm on X is Fréchet smooth, then X has Mazur’s intersection property, [1, Proposition 4.5]. When we do not assume that the dual unit ball is a Corson compact, then the above is no longer true. For example, the Banach space Date: March 16, 2004.