We will establish the following equivariant extension theorem.

Let $G$ be a compact Lie group, $L$ a locally convex metrizable linear $G$-space, and $V$ a closed convex subset of $L$. Denote $G(V):={gv\mid g\in G, v\in V}$ – the equivariant hull of $V$. Then any $G$-equivariant map $f:A\to G(V)$ defined on a closed invariant subset of a metrizable $G$-space $X$, extends to a $G$-equivariant map $F:U\to G(V)$ over some invariant neighborhood $U$ of $A$ in $X$. If, in addition, $V$ contains a $G$-fixed point, the extension can be taken over the whole space, i.e. $U=X$. In particular, any continuous map $f:A\to G(V)$ from a closed subset of a metrizable space $X$, extends to a continuous map $F:U\to G(V)$ over some neighborhood $U$ of $A$ in $X$.
Several applications will be discussed.