= Approximate surjectivity with geometric correction
{title2=$Tx=y,\quad\|x\|\le R\|y\|/(1-k)$}
Let $T:V\to W$ be a <bounded linear operator> with $V$ a <Banach space>. If every $y\in W$ has an approximate preimage $x$ with $\|x\|\le R\|y\|$ and $\|Tx-y\|\le k\|y\|$, where $k<1$, then every $y$ has an exact preimage with the displayed norm bound. Repeatedly correct the residual: the corrections have norms at most $Rk^j\|y\|$ and form an absolutely convergent series; its image under $T$ equals $y$. This proves surjectivity but does not provide a bounded linear right inverse.
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