I would like to prove that $O(n,R)$ is a compact set. Can I just view $O(n,R)$ as a subset of $R^{n*n}$ and prove it is compact by proving it is bounded and closed?
2026-03-25 22:26:03.1774477563
How to prove orthogonal group is compact with induction topology?
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OK.
I’m to prove it’s bounded by using the fact that every number is bounded by 1.
And it’s closed because limit of orthogonal matrix sequence is still orthogonal by continuity of matrix multiplication.