Metric net ($\forall x\in T\exists u\in U:d(x,u)\le \epsilon$)

An $\epsilon$-net of a metric space $(T,d)$ is a subset $U\subseteq T$ such that every $x\in T$ has $d(x,u)\le \epsilon$ for some $u\in U$. Its least possible finite size is the metric covering number with closed balls. A maximal set with pairwise distances greater than $\epsilon$ is an $\epsilon$-net.

Suppose the unit ball $B$ of a $k$-dimensional normed vector space has metric covering number $N(r,B)\le (A/r{)}^k$ at $r=\delta /2$, where $A,\delta >0$. Cover $B$ by at most $(2A/\delta {)}^k$ balls of radius $\delta /2$. Discard every ball missing the unit sphere, and select one point of the unit sphere from each remaining ball. These selected points form a radius-$\delta$ metric net $N_{\delta }$ of the unit sphere, withIndeed, a unit sphere point and the selected point from the same covering ball each lie within $\delta /2$ of its center, so the triangle inequality bounds their distance by $\delta$. The selection ensures that all centers of the new metric net have unit norm; the original covering centers need not. For Euclidean balls, the volumetric bound for Euclidean metric nets supplies the initial metric covering number estimate.

For a symmetric matrix $H$ and an $\eta$-metric net $U$ of the unit sphere, with $0<\eta <1/2$, approximate a maximizing unit vector $x$ by $u\in U$. Expanding $x^{\top }Hx-u^{\top }Hu=(x-u{)}^{\top }Hx+u^{\top }H(x-u)$ bounds its absolute value by $2\eta \Vert H{\Vert }_{\mathrm{op}}$. Rearrangement proves the bound. The volumetric bound for Euclidean metric nets limits the number of needed directions, allowing a union bound to control a random matrix.

A subset of the radius-$R$ Euclidean ball admits an $\epsilon$-metric net of size at most $(1+2R/\epsilon {)}^n$. Take a maximal separated set: its disjoint radius-$\epsilon /2$ Euclidean balls lie in the radius-$(R+\epsilon /2)$ ball. Comparing Lebesgue measures gives the estimate. In particular, the unit sphere admits a net of radius one with at most $3^n$ points.