Define the Banach-Mazur distance between finite-dimensional normed space $X, Y$ as $$d(X,Y)=\inf_{T\in\mathrm{GL}(X,Y)}\|T\|\|T^{-1}\|.$$ Let $\mathrm{BM}(n)$ denote the set of isometry class of $n$-dimensional normed space, then $\log d$ is a distance on $\mathrm{BM}(n)$.
I can understand all statements above, and I know such distance space is call Banach-Mazur compactum, but why does it compact? I think it may follow from the result $$d(X,\ell^2_n)\leq\sqrt{n},$$ but I don't know how to go forward.