Equivariant generalization of Dugundji's theorem (Q1083081)

The author provides an equivariant generalization of Dugundji's extension of Tietze's theorem. The main result of this paper is the following: Theorem. Let G be a compact Lie group, A an invariant closed subset of the metrizable G-space X, and V a convex invariant subset of some locally convex G-space Z. Then each equivariant map \(f: A\to V\) admits an equivariant extension \(F: L\to V\) to some invariant neighbourhood L of the set A in X. The author shows by an example that F cannot be defined on all of X, in general. If, though, the set \(\{\) \(v\in V:\) \(gv=v\) for all \(g\in G\}\) is nonempty, then \(L=X\). Also, if V is a complete set with respect to the natural group of uniformities of the linear topological space Z, then \(L=X\).