Lipschitz equivalent Banach spaces need not be linearly isomorphic

What OpenAI claims

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. (Lean)

Papers (2)

1. Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0

   26 September 2026 · Preprint, not peer reviewed · PDF

   We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to Z⊕_∞ c₀. The same space contains a bi-Lipschitz image of every separable metric space.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.