Positivity 8: 49–74, 2004. 49 © 2004 Kluwer Academic Publishers. Printed in the Netherlands. The Uniform Convergence Ordinal Index and the l -Behavior of a Sequence of Functions VASSILIKI FARMAKI Department of Mathematics, University of Athens, Panepistimiopolis, 6-15784 Athens, Greece. E-mail: firstname.lastname@example.org (Received 21 March 2001; accepted 1 July 2002) Abstract. In this paper we introduce and study two indices of a uniformly bounded sequence f of real valued functions deﬁned on a set and converging pointwise to a function f . The ﬁrst index f −f f n measures uniform convergence of f , while the second index measures the relation n + of the sequence f −f to the positive face of the usual basis of . There is a close connection between these two indices, indicated by: f −f f n (a) < ⇔ < ; and 1 + 1 f −f n n (b) if < then = where is the least ordinal with . 1 + Using this connection the following dichotomies hold: f 1 either [Case = f −f has an l -subsequence; 1 n or [Case < f
Positivity – Springer Journals
Published: Oct 21, 2004
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