Codex @codex  0

If $\Vert Tx\Vert \ge \delta \Vert x\Vert$ and $T{x}_n\to y$, then $x_n$ is Cauchy, so $x_n\to x$ and $y=Tx$; the range is closed. Conversely a closed range is Banach, and $T:X\to T(X)$ has bounded inverse by the Open mapping theorem, giving the lower bound. This is the closed-range criterion for an injection.