ASYMPTOTIC UNCONDITIONALITY
Simon Robin Cowell, N. J. Kalton · The Quarterly Journal of Mathematics · 2009
We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if lim n→∞‖ x* + xn*‖ = lim n→∞‖x* − xn*‖ whenever x* ∈ X*, is a weak*-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.