### Bourgain’s discretization theorem

Bourgain’s discretization theorem asserts that there exists a universal constant $C\in (0,\infty )$ with the following property. Let $X,Y$ be Banach spaces with $dimX=n$. Fix $D\in (1,\infty )$ and set $\delta ={e}^{-{n}^{Cn}}$. Assume that $\mathcal{N}$ is a $\delta $-net in the unit ball of $X$ and that $\mathcal{N}$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...