(a) What is meant by a compact topological group? Explain why $\mathrm{SU}(n)$ is an example of such a group.

[In the following the existence of a Haar measure for any compact Hausdorff topological group may be assumed, if required.]

(b) Let $G$ be any compact Hausdorff topological group. Show that there is a continuous group homomorphism $\rho: G \rightarrow \mathrm{O}(n)$ if and only if $G$ has an $n$-dimensional representation over $\mathbb{R}$. [Here $\mathrm{O}(n)$ denotes the subgroup of $\mathrm{GL}_{n}(\mathbb{R})$ preserving the standard (positive-definite) symmetric bilinear form.]

(c) Explicitly construct such a representation $\rho: \mathrm{SU}(2) \rightarrow \mathrm{SO}(3)$ by showing that $\mathrm{SU}(2)$ acts on the following vector space of matrices,

$$\left\{A=\left(\begin{array}{cc} a & b \\ c & -a \end{array}\right) \in \mathrm{M}_{2}(\mathbb{C}): A+\overline{A^{t}}=0\right\}$$

by conjugation.

Show that

(i) this subspace is isomorphic to $\mathbb{R}^{3}$;

(ii) the trace map $(A, B) \mapsto-\operatorname{tr}(A B)$ induces an invariant positive definite symmetric bilinear form;

(iii) $\rho$ is surjective with kernel $\left\{\pm I_{2}\right\}$. [You may assume, without proof, that $\mathrm{SU}(2)$ is connected.]