Result 324, Functional analysis

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 formalization Disproof or counterexample

The bigger picture

Why it matters

Distances and linear structure need not tell the same story, even in separable Banach spaces. These unreviewed manuscripts report spaces whose distances match within fixed multiplicative factors, but whose vector-space structures cannot be matched by a bounded invertible linear map.

What changes?

The claimed examples are real Banach spaces, complete normed vector spaces, that are separable, meaning they have countable dense subsets. Specifically, a space Z contains no closed subspace linearly isomorphic to c0, the space of real sequences tending to zero with the supremum norm. Yet Z is globally bi-Lipschitz equivalent to Z times c0 with the maximum norm: a bijection distorts all distances by at most fixed multiplicative factors.

What does that help mathematicians do?

The product contains a linear copy of c0, so the claimed absence of such a copy in Z rules out a linear isomorphism between them. This identifies a specific feature of linear structure that global bi-Lipschitz equivalence cannot detect. The second manuscript also reports that every separable metric space embeds bi-Lipschitz into Z, showing that this broad capacity to contain metric spaces can coexist with excluding even a linear copy of c0.

Are there practical applications?

The immediate value is foundational for functional analysis, where researchers compare spaces using both linear maps and distance-preserving estimates. The claimed counterexample sets a limit on that comparison: classification up to bi-Lipschitz equivalence cannot recover linear isomorphism classes of separable real Banach spaces. Additional information would be needed for that stronger classification.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

September 24, 2026 23 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026,
  author = {{OpenAI}},
  title = {{Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf}{OAI:Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026}},
  year = {2026}
}

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

September 26, 2026 25 pages Main result formalized in Lean

We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to Z⊕∞c0Z\oplus_\infty c_0. The same space contains a bi-Lipschitz image of every separable metric space.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026,
  author = {{OpenAI}},
  title = {{Bi-Lipschitz Absorption of $c_0$ Without a Linear Copy of $c_0$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf}{OAI:Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/324.md.

Lipschitz equivalent Banach spaces need not be linearly isomorphic

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized counterexample gives separable real Banach spaces X,YX,Y that are bi-Lipschitz equivalent but not linearly isomorphic. The bijection has lower Lipschitz bound 4/214/21 and upper bound 76/2576/25. The linear obstruction is explicit: XX contains a linear isometric copy of c0(ℓ2)c_0(\ell_2), whereas YY contains no bounded linear copy of that space.

The formalized result constructs one separable real Banach space XX that is bi-Lipschitz equivalent to X×c0X\times c_0 but contains no closed linear subspace isomorphic to c0c_0. The same space contains a bi-Lipschitz copy of c0c_0, is bi-Lipschitz universal for separable metric spaces, and is not linearly isomorphic to X×c0X\times c_0. Thus nonlinear absorption of c0c_0 does not force a linear copy of it.

Comparator links

Result Comparator statement
Bi-Lipschitz equivalent nonisomorphic Banach spaces LipschitzEquivalence.lean
Bi-Lipschitz absorption of c0c_0 C0Absorption.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.