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

If $\Vert Tx\Vert \ge \delta \Vert x\Vert$ and $\Vert S-T\Vert <\delta$, then
$\Vert Sx\Vert \ge (\delta -\Vert S-T\Vert )\Vert x\Vert$. Thus bounded-below maps form an open set.

Surjectivity and Open mapping theorem give $r{B}_Y\subset T(B_X)$. If $\Vert S-T\Vert <r$, iteratively solve with $T$ and correct the error $(S-T)x$; the errors shrink geometrically. Completeness yields an exact preimage under $S$.

For an isomorphism $T$, if $\Vert T^{-1}(S-T)\Vert <1$, thenis invertible by neumann-series perturbation.