Exercise 3, § 6

Let H be a separated pre-Hilbert space. Let $n$ be an integer $\geqslant 1$; denote by $V_n(H)$ the subspace of $H^n$ formed by the orthonormal sequences. Let G be a closed subgroup of $O(n,\mathbf{R})$. a) Prove that one lets the orthogonal group $O(n,\mathbf{R})$ (LIE, IX, §3, n$^o5$, p. 22) operate properly and freely in the space $V_n(H)$ by the formula

b) Let G be a closed subgroup of $O(n,\mathbf{R})$. Prove that the canonical surjection makes $V_n(H)$ a principal fiber space with group G and base $V_n(H)/G$. (Begin by treating the case where $G = O(n,\mathbf{R})$.)

c) Prove that the quotient space $V_n(H)/G$ is metrizable. (Equip $H^n$ with its natural structure of pre-Hilbert space and consider the distance $d$ on $V_n(H)$ given by $d(u, v) =$ inf$_{g\in G}\|u-v\cdot g\|.$)

d) Prove that the quotient space $V_n(H)/G$ is a classifying space for G.

e) Deduce that every compact Lie group possesses a classifying space which is a metrizable topological space. (Apply LIE, IX, §9, n$^o2$, p. 91, th. 1.)