On equivariant embeddings of $G$-spaces

We study a functorial dependence $\tilde{\alpha}$ between maps $h\colon X\to Y$, where $X$ is a $G$-space with continuous action $\alpha$ of the group $G$, and maps $\tilde{\alpha}(h)\colon X\to Y^X$, where $Y^X$ is taken with the compact open topology. The functor $\tilde{\alpha}$ preserves the properties of being one-to-one, of being continuous, of being a topological embedding and, in the case of a compact group, of being a topological embedding with a closed image. For fixed $X$, $\alpha$, and $Y$, the functor $\tilde{\alpha}$ is a topological embedding of $\mathscr C(X,Y)$ into $\mathscr C(X,\mathscr C(G,Y))$. (The topology is compact-open.) If $Y$ is a topological vector space, then $\tilde{\alpha}$ is a monomorphism. If $G$ is locally compact, then there is a continuous action of $G$ on $\mathscr C(G,Y)$ and $\tilde{\alpha}(h)$ is equivariant for any $h$. If $V$ is a locally convex space, then there exists a continuous monomorphism of $G$ into the group of all topological linear transformations of the locally convex space $\mathscr C(G,V)$. For a locally compact group $G$ every completely regular $G$-space can be embedded in a topologically equivariant way in the locally convex space $\mathscr C(G,V)$ under the natural action of the group of all topological linear transformations. (This result was recently obtained by de Vries by means of a different construction.) If $G$ is compact, then the embedding can be made to have a closed image.