Classifications of Baire- functions and -spreading models

Author:
V. Farmaki

Journal:
Trans. Amer. Math. Soc. **345** (1994), 819-831

MSC:
Primary 46B20; Secondary 26A21, 46B15, 46E15

DOI:
https://doi.org/10.1090/S0002-9947-1994-1262339-1

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 of continuous functions, with spreading model of order , , equivalent to the summing basis of , converging pointwise to *f*, then (the index as defined by A. Kechris and A. Louveau). As a corollary of this result we have that the Banach spaces , , which previously defined by the author, consist of functions with rank greater than . For the case we have the equality of these classes. For every countable ordinal number we construct a function *S* with . Defining the notion of null-coefficient sequences of order , , we prove that every bounded sequence of continuous functions converging pointwise to a function *f* with is a null-coefficient sequence of order . As a corollary to this we have the following -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 or is a null-coefficient sequence of order 1.

**[1]**Dale E. Alspach and Spiros Argyros,*Complexity of weakly null sequences*, Dissertationes Math. (Rozprawy Mat.)**321**(1992), 44. MR**1191024****[2]**S. Argyros,*Banach spaces of the type of Tsirelson*(to appear).**[3]**Steven F. Bellenot, Richard Haydon, and Edward Odell,*Quasi-reflexive and tree spaces constructed in the spirit of R. C. James*, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 19–43. MR**983379**, https://doi.org/10.1090/conm/085/983379**[4]**C. Bessaga and A. Pełczyński,*On bases and unconditional convergence of series in Banach spaces*, Studia Math.**17**(1958), 151–164. MR**0115069**, https://doi.org/10.4064/sm-17-2-151-164**[5]**Vassiliki Farmaki,*On Baire-1/4 functions and spreading models*, Mathematika**41**(1994), no. 2, 251–265. MR**1316606**, https://doi.org/10.1112/S0025579300007361**[6]**V. Farmaki and A. Louveau,*On a classification of functions*(unpublished).**[7]**R. Haydon, E. Odell, and H. Rosenthal,*On certain classes of Baire-1 functions with applications to Banach space theory*, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 1–35. MR**1126734**, https://doi.org/10.1007/BFb0090209**[8]**A. S. Kechris and A. Louveau,*A classification of Baire class 1 functions*, Trans. Amer. Math. Soc.**318**(1990), no. 1, 209–236. MR**946424**, https://doi.org/10.1090/S0002-9947-1990-0946424-3**[9]**H. Rosenthal,*A characterization of Banach spaces containing*(to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1262339-1

Article copyright:
© Copyright 1994
American Mathematical Society