Let $\mathfrak{X}$ be a Banach space and $\mathcal{B}(\mathfrak{X})$ the set of bounded linear operators mapping $\mathfrak{X}$ to $\mathfrak{X}$. In [1] below it is shown that the set of invertible operators is not necessarily dense in $\mathcal{B}(\mathfrak{X})$, although it is well known that it is an open set. In [2], it is shown that the set of injections (I suppose we may call this $\mathcal{I}(\mathfrak{X})$) is not necessarily open in $\mathcal{B}(\mathfrak{X})$.
My question is on how "common" injectiveness is in $\mathcal{B}(\mathfrak{X})$. In particular, the results above leave open the possibility that $\mathcal{I}(\mathfrak{X})$ is a dense $G_\delta$ set, and so topologically generic. In either case, it would be interesting to know whether density or the $G_\delta$ condition hold. I believe that the spectral theorem already implies that normal operators on Hilbert spaces may be approximated by injective operators (see for instance [3]). Any results that you guys know of would be appreciated!
[1] Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift
[3] The set of invertible normal operator is dense in the set of normal operator