Compact orbit spaces in Hilbert spaces and limits of edge-colouring models

Let $G$ be a group of orthogonal transformations of a real Hilbert space $H$. Let $R$ and $W$ be bounded $G$-stable subsets of $H$. Let $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_R$ be the seminorm on $H$ defined by $\mathrm{\parallel }x{\mathrm{\parallel }}_R:={\sup }_{r\in R}\mathrm{\mid }⟨r,x⟩\mathrm{\mid }$ for $x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k\mathrm{/}G$ is compact for each $k\in \text{oN}$, then the orbit space $W\mathrm{/}G$ is compact when $W$ is equiped with the norm topology induced by $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_R$. As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lov'asz. It forms the edge-colouring counterpart of the graph limits of Lov'asz and Szegedy, which can be seen as limits of vertex-colouring models. In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called spin models' and vertex models’ respectively.