The Uniform Convergence Ordinal Index and the l1-Behavior of a Sequence of Functions

Abstract

In this paper we introduce and study two indices of a uniformly bounded sequence (f n ) of real valued functions defined on a set Γ and converging pointwise to a function f. The first index ξ (f n ) measures uniform convergence of (f n ), while the second index ξ (f n -f) + measures the relation of the sequence (f n -f) to the positive face of the usual basis of 1 . There is a close connection between these two indices, indicated by: (a) ξ (f n ) < ω 1 ⇔ ξ (f n -f) + < ω 1 ; and (b) if ξ (f n ) < ω 1 then ξ (f n -f) +=ω ζ where ζ is the least ordinal with ξ (f n ) ≤ ω ζ . Using this connection the following dichotomies hold: either [Case ξ (f n ) = ω 1 ](f n -f) has an l 1 +-subsequence; or [Case ξ (f n ) < ω 1 ](f n ) converges weakly to f in the Banach space l∞(Γ). Fixing the least countable ordinal ζ with ξ (f n ) ≤ ω ζ we obtain for every countable ordinal a the further dichotomy: either [Case a < ζ] there exist a subsequence of (f n - f) with l 1 -spreading model of order a; or [Case α ≥ ζ]the sequence (f n ) converges ω α -uniformly to f; equivalently every subsequence of (f n ) has an A ω α-convex block subsequence converging uniformly to f. ((A ξ ) l ≤ ξ<ωl is the complete thin Schreier system introduced previously by the author). There are applications of these results to Banach space theory.