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

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.

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