# Math Help - How to prove O is compact in M?

1. ## How to prove O is compact in M?

Let M be the k*n matrices of rank k(k <= n),O= { A is in M: A^{t}A=I }.
How to prove O is compact in M?

2. I guess you mean compact for a topology which is given by a norm. Since M is a finite dimensional vector space, you have to show that O is closed and bounded.

3. The set of $k\times n$ matrices can by identified in the obvious way with $\mathbb{R}^{nk}$, and can be given the standard metric topology. Heine-Borel applies in this setting, too. Bounded is easy here, and for closed, here's a hint: is the map $A\mapsto AA^T$ continuous?