A weak topology characterization of l₁ (m)

Summary

Abstract. Let m be any cardinal. The main result characterizes ℓ₁ (m) as the only Banach lattice whose positive cone is metrizable in the weak topology. Two related theorems on ℓ₁-sums of Banach spaces are also proved.

Introduction

The three theorems of the paper concern ℓ₁-sums of Banach spaces. The first, which states that the Banach space ℓ_∞(m) contains the ℓ₁-sum of 2^m copies of itself, is essentially a translation into Banach space terms of a theorem of Pondiczery on the product topology [10]. The case m = ℵ₀ of this result is reminiscent of the general theorem [9] that if X is any separable Banach space containing ℓ₁, then X* contains M/(0, 1), the space of finite Borel measures on [0, 1]: the connection resides in the observation that M[0, l] is linearly isomorphic to the ℓ₁-sum of c copies of L₁ (0, 1). This observation is used to prove Theorem 2, which says that if X is as above then X* contains 2^C mutually non-isomorphic closed linear subspaces.

Recall that the unit ball of a Banach space X is metrizable in the weak topology ifX* is separable, and recall too the consequence of the Baire category theorem that if X is infinite-dimensional then the weak topology on X and the weak-star topology on X* are not metrizable. However, there are still some interesting unbounded subsets of Banach spaces that are metrizable in the weak or the weak-star topology. Recall, for example, that the positive cone of the dual of C(K) (that is, the space of continuous functions on a compact metric space K), consisting of the non-negative finite Borel measures on K, is metrizable in the weak-star topology.