Compactness of the set of $n \times n$ orthogonal matrices
Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.
Recall a compact subset of $R^{n \times n}$ is a set that is closed and bounded. One way to show closedness is to observe that the orthogonal matrices are the inverse image of the element $I$ under the continuous map $M \rightarrow MM^T$. Boundedness follows for example from the fact that each column or row is a vector of magnitude $1$.