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Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.

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The column vectors of an orthogonal matrix are unit vectors. And there are $n$ column vectors. –  PEV Jan 23 '11 at 21:28
    
It would also be expeditious to use the operator norm. What is the operator norm of an orthogonal matrix? –  hardmath Jan 23 '11 at 21:42

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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 $1$.

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Can you elaborate on how this implies boundedness? –  learner Dec 12 at 2:38
    
Each entry must be of absolute value at most 1, since the column it is in has magnitude 1, for example. –  Zarrax Dec 12 at 5:02
    
Yes but what is the norm? –  learner Dec 12 at 5:21
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"Magnitude 1" means Euclidean norm here. –  Zarrax Dec 12 at 14:07

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