## A classification of Baire class 1 functions.(English)Zbl 0692.03031

The authors study three variants of ordinal ranks for functions of Baire class 1 and show that for bounded functions these are essentially equivalent. Thus one obtains a hierarchy of Banach algebras $$B_ 1^{\xi}$$ $$(\xi <\omega_ 1)$$ of bounded Baire-class 1 functions with the first level $$B^ 1_ 1$$ corresponding to the strict Baire class 1 functions. Two other approaches to the authors’ classification are given: the first is similar to the Hausdorff-Kuratowski analysis of $$\Delta^ 0_ 2$$ sets via transfinite differences of closed sets and uses alternating sums of usc functions, the second uses a notion of pseudo- uniform convergence. Finally the problem of optimal convergence for derivatives is studied.
Reviewer: K.Gloede

### MSC:

 03E15 Descriptive set theory 46B99 Normed linear spaces and Banach spaces; Banach lattices
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