To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.
PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion.
