Net estimate for a linear operator
= Net estimate for a linear operator
The unit sphere of an $n$-dimensional normed space has a $\delta$-net of size at most $(3/\delta)^n$. If $1-\delta\leq\|Tx\|\leq1+\delta$ on this net and $\delta<1/3$, then
$$
\|T\|\|T^{-1}\|\leq\frac{1+\delta}{1-3\delta}.
$$