On Asymptotic Models in Banach Spaces

A well known application of Ramsey’s Theorem to Banach Space Theory is the notion of a spreading model (ẽi) of a normalized basic sequence (xi) in a Banach space X. We show how to generalize the construction to define a new creature (ei), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem— a strengthened form of Ramsey’s Theorem—to generate asymptotic models with a stronger form of convergence.