Let $E$ be an infinite dimensional Banach space and $F$ a proper subspace of $E$. Let $R: E → F ⊊ E$.
We know that $R$ is linear, injective and that $R(E) ≠ F$ and $R(E)$ is closed.
1) I want to prove that if $R(E) ≠ F$, then there exists a constant $c > 0$ such that $\|Rx\|≥ c \|x\|$. This prove that $R$ is not the limit of a sequence in the linear group $GL(E)$.
2) Now I want to prove that if $\|Rx\|≥ c \|x\|$ for any $x$, $\|T_n^{-1} x\| ≤ c/2 \|x\|$.