TITLE: Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$
QUESTION [5 upvotes]: Is the set of all unitary matrices in $M_2(\mathbb{C})$ is compact? I can show that as determinant map is continuous so unitary matrices are closed but how to show they are bounded? 
Please help.

REPLY [15 votes]: $M_n(\mathbb C)$ is a finite-dimensional space, so it's enough to show that $U_n(\mathbb C)$ is closed and bounded. 
The maps $f_1\colon U\mapsto U^*U-I$ and $f_2\colon U\mapsto UU^*-I$ are continuous since so is the map $U\mapsto U^*$, so $U_n(\mathbb C)=f_1^{-1}(\{0_n\})\cap f_2^{-1}(\{0_n\})$ is closed as an intersection of such two sets.
$U_n(\mathbb C)$ is bounded for the euclidian operator norm, since for each $x$ and $U$ unitary
$$\langle x,x\rangle=\langle U^*Ux,x\rangle=\langle Ux,Ux\rangle$$
(hence $U$ is an isometry, in particular its norm is $1$).