Lipschitz equivalent Banach spaces need not be linearly isomorphic. Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.
There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.
released 2026-09-26 | 1 theorem · 15 lemmas · 21 proofs · 11,057 words |
PLAY LEVEL 2 »(pdf)
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to $Z\oplus_\infty c_0$. The same space contains a bi-Lipschitz image of every separable metric space.