Classification of spaces of Baire functions on ordinal intervals

A complete linear homeomorphic classification of the spaces $B_p[1,\alpha]$ of all Baire functions $f\colon[1,\alpha]\to\mathbb R$ that are defined on ordinal intervals $[1,\alpha]$ and are equipped with the topology of pointwise convergence is given. Since any Baire function on an ordinal interval belongs to the first Baire class, our classification is also a classification of spaces of functions of the first Baire class. A similar classification is given for spaces of two-valued Baire functions $f\colon[1,\alpha]\to\{0,1\}$.