In this paper we prove a
Dugundji Extension Theorem for a large class of monotonically normal spaces,
the generalized ordered spaces. We show that if A is a closed subset of a
generalized ordered space X and if C∗(A) and C∗(X) denote the vector spaces of
continuous, bounded real-valued functions on A and X respectively, then there is a
linear transformation u : C∗(A) → C∗(X) such that for each g ∈ C∗(A), u(g)
extends g and the range of u(g) is contained in the closed convex hull of the
range of g. Furthermore, we give an example which shows that such linear
transformations from C(A), the vector space of all continuous, real-valued
functions on A, to C(X) cannot always be found, even when A is a closed,
separable metrizable subspace of a hereditarily paracompact linearly ordered
space.
