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.