I have already shown that $O(n), SO(n), U(n), SU(n)$ and $Sp(n)$ are closed. Now I want to show that they are bounded. But when I tried, I noticed I need a metric or a norm on these sets. But there are several possibibilities to define a norm on these sets.
(1) What's the norm that is usually put on these matrix groups?
(2) If we just "linearise" and glue one column after another to get an $n^2 \times 1$ vector and endow this with the Euclidean metric, would it be ok to argue that since the matrices are orthogonal, each column has (Euclidean) length $1$ and therefore the $n^2 \times 1$ vector has at most length $n$ hence $O(n)$ etc. are contained in the (closed) ball $B(0,n)$?