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 (i.e. the space of differences of bounded semicontinuous functions on ) and especially of (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 belongs to if and only if the -oscillation, , of is bounded and in this case . Also, we classify into a decreasing family of Banach spaces whose intersection is equal to and . These spaces are characterized by spreading models of order equivalent to the summing basis of , and for every function in it is valid that is bounded. Finally, using the notion of null-coefficient of order sequence, we characterize the Baire-1 functions not belonging to .

**[A-A]**Dale E. Alspach and Spiros Argyros,*Complexity of weakly null sequences*, Dissertationes Math. (Rozprawy Mat.)**321**(1992), 44. MR**1191024****[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 𝑐₀-spreading models*, Trans. Amer. Math. Soc.**345**(1994), no. 2, 819–831. MR**1262339**, 10.1090/S0002-9947-1994-1262339-1**[F-L]**V. Farmaki and A. Louveau,*On a classification of functions*, (unpublished).**[H-O-R]**R. Haydon, E. Odell, and H. Rosenthal,*On certain classes of Baire-1 functions with applications to Banach space theory*, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 1–35. MR**1126734**, 10.1007/BFb0090209**[K-L]**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**, 10.1090/S0002-9947-1990-0946424-3**[R1]**Haskell Rosenthal,*A characterization of Banach spaces containing 𝑐₀*, J. Amer. Math. Soc.**7**(1994), no. 3, 707–748. MR**1242455**, 10.1090/S0894-0347-1994-1242455-4**[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

Email:
vgeorgil@atlas.uoa.gr

DOI:
http://dx.doi.org/10.1090/S0002-9947-96-01719-9

Received by editor(s):
August 22, 1994

Article copyright:
© Copyright 1996
American Mathematical Society