Linear extensions of some Baire-one functions

Abstract

Let X be a Hausdorff topological space, and let {mathscr {B}}_1(X) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let {mathscr {F}}(A) be the set of functions Arightarrow {mathbb R} with a property {mathscr {F}} making {mathscr {F}}(A) a linear subspace of {mathscr {B}}_1(A). We give a sufficient condition for the existence of a linear extension operator T_A:{mathscr {F}}(A)rightarrow {mathscr {F}}(X), where {mathscr {F}} means to be piecewise continuous on a sequence of closed andG_delta subsets ofX and is denoted by {mathscr {P}_0}. We show that T_A restricted to bounded elements of {mathscr {F}}(A) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.