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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 $B(0,n)$?

learner
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  • You could go with the operator norm. More to the point, it doesn't matter for this question because all norms are equivalent: $|A|_1 \le C |A|_2$ for any two norms. –  Dec 12 '14 at 02:55
  • re: (2), you absolutely could do it that way (this is often called the Frobenius norm, and in fact, it's length is at most $\sqrt n$), but as @ChristianRemling notes, you can use any convenient norm, and the operator norm makes quick work of all 5 groups. – BaronVT Dec 12 '14 at 03:02
  • @BaronVT But Wikipedia states that "These are different from the entrywise $p$-norms" suggesting that not all matrix norms are equivalent. So I cannot use any norm, I have to know which norm is used commonly on matrix groups. – learner Dec 12 '14 at 03:09
  • The caution there is that $|A|_1$ could mean the induced $1$-norm, or the entrywise $1$-norm, and in general you'll get different values (so be careful of notation). The important thing is that all of these norms (induced, entrywise, etc.) satisfy the definition of a norm (positivity, absolute scalability, triangle inequality), and all norms are equivalent (they don't give you the same value, per se, but $c_1|A|_1 \leq |A|_2 \leq c_2 |A|_1$ for any $A$, thus they induce the same topology, in particular the bounded sets are the same). – BaronVT Dec 12 '14 at 03:17
  • If you're interested in learning more about these norms, and seeing what the constants of equivalency are, I suggest Horn and Johnson's Matrix Analysis. – BaronVT Dec 12 '14 at 03:21
  • And these $p$-norms are also equivalent to the operator norm? – learner Dec 12 '14 at 04:22

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