FLAT SETS, ℓ_p-GENERATING AND FIXING c₀ IN THE NONSEPARABLE SETTING

Abstract

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak * Kadec-Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c 0 (ω 1 )-generated or p (ω 1 )-generated, where p ∈ (1, ∞). In particular, we show that every subspace of c 0 (ω 1 ) is c 0 (ω 1 )-generated and every subspace of p (ω 1 ) is p (ω 1 )-generated for every p ∈ (1, ∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal's theorem on fixing c 0 (ω 1 ).2000 Mathematics subject classification: primary 46B26; secondary 46B03, 46B20.