= Net estimate for a linear operator
Let $E$ be an $n$-dimensional <normed vector space>, let $0<\delta<1/3$, and let $T:E\to F$ be a <linear operator> into another <normed vector space>. A maximal $\delta$-separated subset of the unit <sphere> is a $\delta$-net. Disjoint balls of radius $\delta/2$ about its points lie in the ball of radius $1+\delta/2$, so comparison of <volumes> bounds its size by $(1+2/\delta)^n\leq(3/\delta)^n$.
Suppose $1-\delta\leq\|Tv\|\leq1+\delta$ at every net point $v$. Approximating an arbitrary unit <vector> $x$ by $v$ gives $\|Tx\|\leq1+\delta+\delta\|T\|$, hence $\|T\|\leq(1+\delta)/(1-\delta)$. The reverse <triangle inequality> gives
$$
\|Tx\|\geq1-\delta-\delta\|T\|\geq\frac{1-3\delta}{1-\delta}>0.
$$
Thus $T$ is injective, and its inverse on its image satisfies
$$
\|T\|\|T^{-1}\|\leq\frac{1+\delta}{1-3\delta}.
$$
This turns finitely many norm estimates into uniform control of an entire finite-dimensional <linear subspace>, as used in random <metric embeddings>.
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