= Geometric correction for approximate surjectivity
{title2=$\|Tx-y\|\le k\|y\|\quad\Longrightarrow\quad\|x_{\rm exact}\|\le R\|y\|/(1-k)$}
Let $T:E\to F$ be a <bounded linear operator>, $E$ a <Banach space>, and $F$ a <normed vector space>. Suppose each residual $r$ has an approximate preimage of norm at most $R\|r\|$ leaving error at most $k\|r\|$, where $0<k<1$. Correct successive residuals to get vectors $x_j$ with $\|x_j\|\le Rk^j\|y\|$. Their absolutely convergent sum in $E$ maps exactly to $y$, and its norm is at most $R\|y\|/(1-k)$. Completeness of $F$ is not needed for this argument.
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