The Banach-Mazur metric is a distance on the space of all isomorphic Banach spaces. If $B_{1},B_{2}$ are $n$-dimensional Banach spaces, the distance between them is

Then $d$ satisfies the triangle inequality, and $d(B_{1},B_{2})=0$ if and only if $B_{1}$ and $B_{2}$ are isometric. The space of isometry classes of $n$-dimensional Banach spaces under this metric is a compact metric space, known as a Banach-Mazur compactum.