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As part of a problem I've been set, I'm required to show that if $G$ is a compact group then there is a continuous group homomorphism $G \to O(n)$ if and only if $G$ has an $n$-dimensional (continuous) representation over $\mathbb{R}$.

One direction is easy: if $\rho : G \to O(n)$ is a continuous group homomorphism then it is a representation since $O(n) \le GL_n(\mathbb{R})$.

The converse is less easy (I think), and I can't really see where to begin. I'd like to find some continuous group homomorphism $\theta : GL_n(\mathbb{R}) \to O(n)$, but I'm not having much luck.

Any insight would be much appreciated.

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    $\begingroup$ Do you remember how this was done for G finite? Can we do a similar thing...? $\endgroup$
    – Conor
    May 12, 2012 at 19:23

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Okay, so I imagine you've done this by now, but just in case, and for the sake of future readers...

One definition of the orthogonal group $O(n)$ is the set of matrices preserving the inner product on $\mathbb R^n$. Or, well, an inner product: if you can find an inner product that is preserved by your matrix representations, those matrices will be orthogonal with respect to it. But Weyl's unitary trick gives us exactly such an inner product!

There are some details to be worked out, but that's the core of it, I believe.

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