Baire classes of affine vector-valued functions

TL;DR

This paper explores the Baire classification of affine vector-valued functions in Fréchet spaces, linking the classification to the approximation property and extending scalar results to vector-valued contexts.

Contribution

It extends scalar Baire class results to vector-valued affine functions, relates the first Baire class to the approximation property, and establishes an affine Jayne-Rogers selection theorem.

Findings

Vector-valued Mokobodzki's result depends on the approximation property.

Extended scalar affine class results to vector-valued functions.

Established an affine version of the Jayne-Rogers selection theorem.

Abstract

We investigate Baire classes of strongly affine mappings with values in Fr\'echet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki's result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of $L_1$-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the abstract Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of these results have weaker conclusions than their scalar versions. We also establish an affine version of the Jayne-Rogers selection theorem.