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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Classifications of Baire-$ 1$ functions and $ c\sb 0$-spreading models


Author: V. Farmaki
Journal: Trans. Amer. Math. Soc. 345 (1994), 819-831
MSC: Primary 46B20; Secondary 26A21, 46B15, 46E15
MathSciNet review: 1262339
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Abstract: 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 \leq \xi < {\omega _1}$, equivalent to the summing basis of $ {c_0}$, converging pointwise to f, then $ {r_{{\text{ND}}}}(f) > {\omega ^\xi }$ (the index $ {r_{{\text{ND}}}}$ as defined by A. Kechris and A. Louveau). As a corollary of this result we have that the Banach spaces $ {V_\xi }(K)$, $ 1 \leq \xi < {\omega _1}$, 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 countable ordinal number $ \xi $ we construct a function S with $ {r_{{\text{ND}}}}(S) > {\omega ^\xi }$. Defining the notion of null-coefficient sequences of order $ \xi $, $ 1 \leq \xi < {\omega _1}$, we prove that every bounded sequence $ ({f_n})$ of continuous functions converging pointwise to a function f with $ {r_{{\text{ND}}}}(f) \leq {\omega ^\xi }$ is a null-coefficient sequence of order $ \xi $. As a corollary to this we have the following $ {c_0}$-spreading model theorem: Every nontrivial, weak-Cauchy sequence in a Banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of $ {c_0}$ or is a null-coefficient sequence of order 1.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1262339-1
PII: S 0002-9947(1994)1262339-1
Article copyright: © Copyright 1994 American Mathematical Society