Exercise 12, § 3

Let $K$ be a commutative field, $E$ a vector space over $K$, $F$ a linear subspace of $E$, $p$ the canonical homomorphism of $E$ onto $E/F$, and $A$ the set of homomorphisms $f : E/F \to E$ such that $p \circ f$ is the identity homomorphism of $E/F$.

a) If $f \in A$ and $h \in \mathrm{Hom}(E/F, F)$, then $f + h \in A$. Show that if $f, f' \in A$, there exists one and only one $h \in \mathrm{Hom}(E/F, F)$ such that $f + h = f'$. One can therefore regard $A$ as an affine space of which $\mathrm{Hom}(E/F, F)$ is the space of translations.

b) Let $B$ be the set of linear subspaces of $E$ that are supplementary to $F$ in $E$. Show that $f \mapsto f(E/F)$ is a bijection of $A$ onto $B$. One can therefore regard $B$ as an affine space of which $\mathrm{Hom}(E/F, F)$ is the space of translations.

c) Show that if $u$ is an automorphism of $E$ such that $u(F) = F$, then the mapping $f \mapsto u \circ f$ is an affine bijection of $A$ onto $A$. (Choose an origin in $A$, and observe that the mapping $h \mapsto u \circ h$ is an automorphism of the vector space $\mathrm{Hom}(E/F, F)$.) From this, deduce that the mapping $F' \mapsto u(F')$ is an affine bijection of $B$ onto $B$.

d) Assume that $K = \mathbf{R}$ and $\dim E < +\infty$. Let $G$ be a compact group, and $\rho$ a continuous linear representation of $G$ in $E$ such that $\rho(s)(F) = F$ for all $s \in G$. Show that there exists an $F' \in B$ such that $\rho(s)(F') = F'$ for all $s \in G$. (Make use of c) and Lemma 2 of No. 2.) Obtain this result anew using Prop. 1 of No. 1.