Theorem 2 (Ryll-Nardzewski)

— Let E be a normed space and K a non-empty convex subset of E, which is compact for the weakened topology $\sigma(E, E')$. Let G be a group of isometric affine transformations of K. Then there exists a point in K which is invariant under G.

For every $g \in G$, let $K_g$ denote the set of all points x in K such that $gx = x$; let K be assigned the weakened topology; each set $K_g$ is convex and closed in the compact space K. We shall prove that the intersection $\bigcap_{g \in G} K_g$ is non-empty; for this, it is enough to prove that the set $K_{g_1} \cap ... \cap K_{g_n}$ is non-empty for every $g_1, ..., g_n$ in G. Fix $g_1, ..., g_n$ and let H be the subgroup of G generated by $\{ g_1, ..., g_n \}$. Choose a point a in K and let L denote the closed convex envelope of the orbit Ha of a. Let D be the countable set of elements of the form $\lambda_1 h_1 a + \cdots + \lambda_m h_m a$, where $\lambda_1, ..., \lambda_m$ are positive rational numbers such that $\lambda_1 + \cdots + \lambda_m = 1$, and $h_1, ..., h_m$ are elements in H. The closure $\overline{D}$ of D for the strong topology, is convex, hence it is closed for $\sigma(E, E')$ (IV, p. 4, prop. 2); therefore $\overline{D} = L$ and this proves that L is a metric space satisfying the first axiom of countability for the distance $(x, y) \mapsto \|x - y\|$. We can now apply lemma 2. There exists a point b in L which is invariant under H, hence $b \in K_{g_1} \cap ... \cap K_{g_n}$.