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Let $A$ be a matrix in $SL_2(\mathbb R)$. Define the trace norm to be

$$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$

Does this give a continuous map from $SL_2(\mathbb R)$, or maybe some bigger group, to $\mathbb R$? Does the image of $SL_2(\mathbb R)$ avoid $0$?

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Okay, I removed my comments, as promised earlier. The point essentially is this: 1. All norms on the vector space $\mathbb{R}^N$ are equivalent, hence they are all continuous with respect to each other. 2. What you're writing is a norm (that's the hard thing to check!) on $\operatorname{Mat}_{2 \times t}(\mathbb{R})$ (or $\mathbb{R}^{2\times2}$ if you prefer). 3. The restriction of a continuous map to a subset is continuous. –  t.b. Sep 17 '11 at 9:57

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