A linear extension theorem
E. Michael, Aleksander Pełczyński · Illinois Journal of Mathematics · 1967
IntroductionLet T be a topological space, S closed subset of T, and C(S) and C(T) the Banach spaces of bounded, continuous complex (or real) functions on S and T, respectively.Let E C(S) and H C(T) be closed subspaces.A continuous linear mp u E--H is called a linear extension if u(f) is an extension of f for every f E. The purpose of this paper is to study the exist- ence of linear extensions of norm one.If H C(T), our problem was completely settled by Borsuk [3] for separable metric T, and subsequently by Dugundji [6, Theorem 5] for all metric T. THEOREM 1.1 (Borsuk-Dugundji).If T is metrizable, there exists a linear extension u C( S) C(T) of norm one.If H is a proper subspace of C(T), the situation becomes more complicated, and Example 9.2 shows that no linear extension u C(S) --H need exist even when every f e C(S) cn be extended to some f' e H.We therefore introduce the following concept: DEFINITION 1.2.The pair (E, H) has the bounded extension property if, given any e > 0, every f E has a bounded family of extensions {f,: W S, WopeninT} H such that f.(x) -< e whenever x e T W.Note that the pair (C(S), C(T) has this property whenever T is normal.The following result was proved by the second author in [13] and [14].THEOREM 1.3.If T is compact metric, and if (C(S), H) has the bounded extension property, then there exists a linear extension u C(S) --H of norm o?e.Perhaps the most interesting application of Theorem 1.3 was to the case where T is the unit circle in the complex plane, H C(T) is the disc algebra (i.e.H consists of boundary values of continuous functions on the unit disc