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. (Lean)

Manuscripts (2)

Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026}

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.

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

September 26, 2026 · source · Lean formalization available · cite as \cite{OAI:Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026}

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.

Result 324 · Functional analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.