Mazur-Ulam Theorem With Gromov-Hausdorff Distance

It is shown that two Banach spaces are linearly isometric if and only if the Gromov–Hausdorff distance between them is finite, in particular, zero. The proof is compilative and relies on results obtained by many researchers on the approximability of almost-surjective almost-isometries by linear surjective isometries. In the finite-dimensional case, previously obtained by I.~Mikhailov, a simpler proof under weaker assumptions is given. In the finite-dimensional case, a criterion for isometry in terms of finite (compact) subsets is also given.