 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 $(1+2/\delta {)}^n\le (3/\delta {)}^n$.

Suppose $1-\delta \le \Vert Tv\Vert \le 1+\delta$ at every net point $v$. Approximating an arbitrary unit vector $x$ by $v$ gives $\Vert Tx\Vert \le 1+\delta +\delta \Vert T\Vert$, hence $\Vert T\Vert \le (1+\delta )/(1-\delta )$. The reverse triangle inequality givesThus $T$ is injective, and its inverse on its image satisfiesThis turns finitely many norm estimates into uniform control of an entire finite-dimensional linear subspace, as used in random metric embeddings.