P. Kiriakouli
Characterizations of spreading models of $l^{1}$

Comment.Math.Univ.Carolinae 41,1 (2000) 79-95.

Abstract:Rosenthal in [11] proved that if $(f_{k})$ is a uniformly bounded sequence of real-valued functions which has no pointwise converging subsequence then $(f_{k})$ has a subsequence which is equivalent to the unit basis of $l^{1}$ in the supremum norm. \par Kechris and Louveau in [6] classified the pointwise convergent sequences of continuous real-valued functions, which are defined on a compact metric space, by the aid of a countable ordinal index ``$\gamma $''. In this paper we prove some local analogues of the above Rosenthal 's theorem (spreading models of $l^{1}$) for a uniformly bounded and pointwise convergent sequence $(f_{k})$ of continuous real-valued functions on a compact metric space for which there exists a countable ordinal $\xi $ such that $\gamma ((f_{n_{k}}))> \omega ^{\xi }$ for every strictly increasing sequence $(n_{k})$ of natural numbers. Also we obtain a characterization of some subclasses of Baire-1 functions by the aid of spreading models of $l^{1}$.

Keywords: uniformly bounded sequences of continuous real-valued functions, convergence index, spreading models of $l^{1}$, Baire-1 functions
AMS Subject Classification: 46B20, 46E99