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Let $\mathrm{M}(n,\mathbb{R})$ be the space of $n\times n$ matrices over $\mathbb{R}$. Consider the function

$m \in \mathrm{M}(n,\mathbb{R}) \mapsto (a_{11},\dots,a_{1n},a_{21}\dots,a_{nn}) \in \mathbb{R}^{n^2}$.

The space $\mathrm{M}(n,\mathbb{R})$ is locally Euclidean at any point and we have a single chart atlas. I read that the function is bicontinuous, but what is the topology on $\mathrm{M}(n,\mathbb{R})$?

Second question... in what sense it is defined a $C^{\infty}$ structure when there are no (non-trivial) coordinate changes? Do we have to consider just the identity change?


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up vote 4 down vote accepted

The topology on $\mathrm{M}(n, \mathbb{R})$ making the map you give bicontinuous is, well, the topology that makes that map bicontinuous. Less tautologically, it is the topology induced by any norm on $\mathrm{M}(n, \mathbb{R})$. (Recall that all norms on a finite-dimensional real vector space are equivalent.)

Because Euclidean space has a canonical smooth structure, the fact that you have a single-chart atlas means that you can give $\mathrm{M}(n, \mathbb{R})$ the smooth structure by pulling it back from $\mathbb{R}^{n^2}$. There are no compatibility conditions to verify: the identity transition map will always be smooth.

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