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Transactions of the American Mathematical Society

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On Baire-1/4 functions

Author: Vassiliki Farmaki
Journal: Trans. Amer. Math. Soc. 348 (1996), 4023-4041
MSC (1991): Primary 46B03; Secondary 46B25
MathSciNet review: 1373633
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Abstract: We give descriptions of the spaces $D(K)$ (i.e. the space of differences of bounded semicontinuous functions on $K$) and especially of $B_{1/4}(K)$ (defined by Haydon, Odell and Rosenthal) as well as for the norms which are defined on them. For example, it is proved that a bounded function on a metric space $K$ belongs to $B_{1/4}(K)$ if and only if the $\omega ^{ % \mathrm {th}}$-oscillation, $\mathrm {osc}_{\omega }f$, of $f$ is bounded and in this case $\| f\|_{1/4}=\|\, |f|+ \widetilde {\mathrm {osc}}_{\omega } f\|_{\infty }$. Also, we classify $B_{1/4}(K)$ into a decreasing family $(S_{\xi }(K))_{1\leq \xi <\omega _1}$ of Banach spaces whose intersection is equal to $D(K)$ and $S_1 (K)=B_{1/4}(K)$. These spaces are characterized by spreading models of order $\xi $ equivalent to the summing basis of $c_0$, and for every function $f$ in $S_{\xi }(K)$ it is valid that $\mathrm {osc}_{\omega ^{\xi }}f$ is bounded. Finally, using the notion of null-coefficient of order $\xi $ sequence, we characterize the Baire-1 functions not belonging to $S_{\xi }(K)$.

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  • [A-A] D. Alspach and S. Argyros, Complexity of weally null sequences, Dissertationes Mathematicae 321 (1992), 44 pp. MR 93j:46014
  • [C-M-R] F. Chaatit, V. Mascioni and H. Rosenthal, On functions of finite Baire index, (to appear).
  • [F1] V. Farmaki, On Baire-1/4 functions and spreading models, Mathematika 41 (1994), 251-265. CMP 95:08
  • [F2] V. Farmaki, Classifications of Baire-1 functions and $c_0$-spreading model, Trans. Amer. Math. Soc. 345 (1994), 819--831. MR 96c:46017
  • [F-L] V. Farmaki and A. Louveau, On a classification of functions, (unpublished).
  • [H-O-R] R. Haydon, E. Odell and H. Rosenthal, On certain classes of Baire-1 functions with applications to Banach space theory, Springer-Verlag Lecture Notes 1470 (1991), 1-35. MR 92h:46018
  • [K-L] A.S. Kechris and A. Louveau, A classification of Baire class 1 functions, Trans. Amer. Math. Soc. 318 (1990), 209-236. MR 90f:26005
  • [R1] H. Rosenthal, A characterization of Banach spaces containing $c_{0}$, J.A.M.S. 7 (1994), 707--745. MR 94i:46032
  • [R2] H. Rosenthal, Differences of bounded semicontinuous functions I, (to appear)

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Additional Information

Vassiliki Farmaki
Affiliation: Department of Mathematics, Panepistimiopolis, 15784, Athens, Greece

Received by editor(s): August 22, 1994
Article copyright: © Copyright 1996 American Mathematical Society

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