Exercise 40, § 9

Let H be a finite-dimensional real Lie group and $H_0$ its identity component.

(a) Suppose that $H_0$ satisfies condition (F) of Exercise 38, that $H/H_0$ is finite and that the centre of $H_0$ is compact. Let G be a finite-dimensional real Lie group and $f : H \to G$ a continuous homomorphism with discrete kernel. Then $f(H)$ is closed in G. (Reduce it to the case where H is connected. As the kernel of f is discrete, the centre of $f(H)$ is the image under f of the centre of H (Lemma 1) and hence is closed in G. Apply Exercise 39 (c).)

(b) Suppose that $H_0$ is semi-simple and that $H/H_0$ is finite. Then the image of H under a finite-dimensional continuous linear representation is closed in the linear group. (Apply (a) and Exercise 7 (b).)