CLASSIFICATIONS OF BAIRE-1 FUNCTIONS AND C(O)-SPREADING MODELS

We prove that if for a bounded function f defined on a compact space K there exists a bounded sequence (f(n)) of continuous functions, with spreading model of order xi, 1 less-than-or-equal-to xi < omega1, equi valent to the summing basis of c0, converging pointwise to f, then r(N D)(f) > omega(xi) (the index r(ND) as defined by A. Kechris and A. Lou veau). As a corollary of this result we have that the Banach spaces V( xi)(K), 1 less-than-or-equal-to xi < omega1, which previously defined by the author, consist of functions with rank greater than omega(xi). For the case xi = 1 we have the equality of these classes. For every c ountable ordinal number xi we construct a function S with r(ND)(S) > o mega(xi). Defining the notion of null-coefficient sequences of order x i, 1 less-than-or-equal-to xi < omega1, we prove that every bounded se quence (f(n)) of continuous functions converging pointwise to a functi on f with r(ND)(f) less-than-or-equal-to omega(xi) is a null-coefficie nt sequence of order xi. As a corollary to this we have the following c0-spreading model theorem: Every nontrivial, weak-Cauchy sequence in a Banach space either has a convex block subsequence generating a spre ading model equivalent to the summing basis of c0 or is a null-coeffic ient sequence of order 1.