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Denote by $\mathcal L'(\mathrm R^n,\mathrm R^m)$ and $\mathcal L_\prime (\mathrm R^n,\mathrm R^m)$ the subsets formed by the surjective and the injective mappings, respectively, of the normed space $\mathcal L(\mathrm R^n,\mathrm R^m)$ (whose vectors are the linear mappings from $\mathrm R^n$ to $\mathrm R^m$).

I'm trying to prove the following:

  • If $m \leq n$, then $\mathcal L'(\mathrm R^n,\mathrm R^m)$ is dense in $\mathcal L(\mathrm R^n,\mathrm R^m)$;

  • If $n \leq m$, then $\mathcal L_\prime (\mathrm R^n,\mathrm R^m)$ is dense in $\mathcal L(\mathrm R^n,\mathrm R^m)$.

I have already proved that $\mathcal L'(\mathrm R^n,\mathrm R^m)$ and $\mathcal L_\prime (\mathrm R^n,\mathrm R^m)$ are open subsets (regardless of whether $m \leq n$ or $n \leq m$), using continuity of determinant and adequate restrictions/extensions of domain. However, the trick does not seem to work in the present case.

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    I think you should add a "analysis" tag since you need the concept of "limit"?2011-08-05

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