  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 $\|.\|_R$ be the
seminorm on $H$ defined by $\|x\|_R:=\sup_{r\in R}|\langle r,x\rangle|$ for
$x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is
compact for each $k\in\oN$, then the orbit space $W/G$ is compact when $W$ is
equiped with the norm topology induced by $\|.\|_R$.
  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.
