If $\|Tx\|\ge\delta\|x\|$ and $Tx_n\to y$, 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 (functional analysis)>, giving the lower bound. This is the <closed-range criterion for an injection>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 22I / b / Solution

 Solution (source code)