# Lipschitz equivalent Banach spaces need not be linearly isomorphic > Two separable Banach spaces match distances up to fixed factors but are not linearly the same, answering Kalton. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-324/ Last updated: 2026-10-14 **Two separable Banach spaces match distances up to fixed factors but are not linearly the same, answering Kalton.** **Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 24, 2026 · **Field:** Functional analysis · **Entry:** 324 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A bi-Lipschitz map stretches or shrinks every distance by at most fixed factors, but need not respect addition. A long-standing question asked whether such a map between two separable Banach spaces forces them to be linearly the same. The first paper proves the answer is no, with explicit factors 4/21 and 76/25, and Lean checks it. The second paper proves a related result: a space that swallows c₀ nonlinearly but contains no linear copy of it. Lean checks that too. ## At a glance - **First step:** 1978 (Israel Aharoni and Joram Lindenstrauss, nonseparable examples and the question for separable spaces) - **Significance:** Major. It answers two problems from Kalton's 2008 survey that had stayed open for separable Banach spaces. - **Trust:** Formal proof listed - **Field:** Functional analysis (2 papers) ## The explanation Take the map f(x) = x + sin(x)/2 on the real line. Its slope always lies between 1/2 and 3/2. So f never shrinks a distance below half, or stretches it beyond one and a half times. It sends 0 to 0 and π to π. But it sends π/2 to about 2.071, not to π/2 ≈ 1.571. So f is not linear, yet it controls every distance. A map like this, onto and with two-sided distance bounds, is called bi-Lipschitz. A Banach space is a vector space with a length for each vector, where good sequences always have limits. It is separable if a countable set comes arbitrarily close to every point. Two Banach spaces are linearly isomorphic if a linear map matches them with bounded distortion. Every linear isomorphism is bi-Lipschitz. The question is the reverse. If a bi-Lipschitz map exists, must a linear isomorphism exist too? Aharoni and Lindenstrauss built nonseparable counterexamples and asked for separable ones[1]. Ribe later built separable spaces that are uniformly homeomorphic but not isomorphic[2]. That is a weaker kind of matching, with no fixed distance factors. Godefroy, Kalton and Lancien proved a rigidity result. Any Banach space bi-Lipschitz equivalent to c₀ is linearly isomorphic to c₀[3]. Here c₀ is the space of sequences that tend to zero, measured by their largest entry. In the wider quasi-Banach setting, Albiac and Kalton found separable counterexamples for every 0 < p < 1[4]. Kalton listed the separable question as Problem 3 in his 2008 survey[5]. The first paper answers the question negatively for separable Banach spaces[6]. It builds spaces X and Y and a bijection between them. Every distance changes by a factor between 4/21 and 76/25. In decimals that is between about 0.190 and 3.04. Two points 10 apart in X land between about 1.905 and 30.4 apart in Y. Yet X contains a copy of c₀(ℓ₂), a c₀ of Hilbert spaces, while Y contains none[7]. The paper notes this does not rule out an ordinary c₀ inside Y[8]. It also gives no complex-scalar or reflexive version[9]. The obstacle is that a bi-Lipschitz map ignores addition, yet the conclusion is linear. The paper bridges this with Lipschitz-free spaces. These turn a distance-preserving picture into a linear one. A careful geometric condition makes a key linear map completely continuous. Such a map sends every weakly null sequence to a norm-null one[10]. That property blocks any copy of c₀(ℓ₂) in Y. The second paper treats Kalton's Problem 2[11]. It builds a separable space Z with no linear copy of c₀. Yet Z is bi-Lipschitz equivalent to Z combined with an extra copy of c₀[12]. So Z contains c₀ in the bi-Lipschitz sense, but not linearly. Aharoni proved that every separable metric space embeds bi-Lipschitzly into c₀[13]. So the same Z contains a bi-Lipschitz copy of every separable metric space[14]. How sure is this? Lean, a computer proof checker, proves both main results. It covers the pair X and Y with the bounds 4/21 and 76/25, and the missing copy of c₀(ℓ₂). It also covers the space Z that absorbs c₀ without a linear copy. No specialist review has been published, but both statements have Lean proofs. **What it doesn't show.** This does not give complex-scalar or reflexive examples, and it does not say the space Y lacks an ordinary copy of c₀. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major. Both main theorems have Lean proofs; they answer two of Kalton's 2008 problems for separable real Banach spaces. ## Words used - **Banach space:** A vector space with a notion of length in which every Cauchy sequence has a limit. - **Separable:** Having a countable set of points that comes arbitrarily close to every point. - **Bi-Lipschitz map:** A map that changes every distance by a factor between two fixed positive numbers. - **Linearly isomorphic:** Matched by a linear map that is bounded with a bounded inverse. - **c₀:** The space of real sequences that tend to zero, with length given by the largest entry. - **c₀(ℓ₂):** Sequences of vectors from the Hilbert space ℓ₂ whose lengths tend to zero. - **Lipschitz-free space:** A Banach space built from a metric space so that Lipschitz maps become linear maps. - **Completely continuous map:** A linear map that turns every weakly null sequence into a sequence whose lengths tend to zero. ## The original papers (2) - [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf) - September 24, 2026 · Formal proof listed - [Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf) - September 26, 2026 · Formal proof listed - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/324.md) ## Quoted lines from the paper (14) [1] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > At the same time, Aharoni and Lindenstrauss constructed Lipschitz equivalent, nonisomorphic nonseparable Banach spaces and explicitly asked for separable examples. [2] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > Ribe later constructed separable uniformly homeomorphic, nonisomorphic spaces. [3] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > Godefroy, Kalton and Lancien proved that a Banach space Lipschitz isomorphic to \(c_0\) is linearly isomorphic to \(c_0\), and that being linearly isomorphic to a subspace of \(c_0\) is invariant under Lipschitz isomorphism. [4] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > In the quasi-Banach setting, Albiac and Kalton obtained separable counterexamples for every \(0 Kalton recorded the separable Lipschitz-isomorphism question as Problem 3 in his 2008 survey; Albiac, Ansorena, B\'{i}ma and C\'{u}th still described the Banach case as open in the first paragraph of their paper published online in December 2025. [6] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem. [7] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > The space \(X\) contains a linearly isometric copy of \(c_0(\ell_2)\), while \(Y\) contains no closed subspace linearly isomorphic to \(c_0(\ell_2)\). [8] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > It does not say that \(Y\) contains no ordinary \(c_0\). [9] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > It asserts neither reflexivity nor a complex-scalar analogue. [10] From [Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic](https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf): > Section … proves that local orthogonality makes \(q\) \emph{completely continuous}: a weakly null sequence in \(E\) has norm-null image. [11] From [Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf): > Kalton recorded the latter question as Problem 2, separately from the separable Lipschitz-isomorphism question in Problem 3. [12] From [Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf): > We construct a separable real Banach space \(Z\) that contains no linear copy of \(c_0\), yet is bi-Lipschitz equivalent to \(Z\oplus_\infty c_0\). [13] From [Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf): > Aharoni proved that every separable metric space embeds bi-Lipschitzly into \(c_0\); composition therefore makes this one space \(Z\) a bi-Lipschitz universal target for separable metric spaces. [14] From [Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf): > The same space contains a bi-Lipschitz image of every separable metric space. ## Other sources - [openai/math, Lean scope note for family 324](https://github.com/openai/math/blob/main/lean/docs/324.md): “The formalized counterexample gives separable real Banach spaces $X,Y$ that are bi-Lipschitz equivalent but not linearly isomorphic.” - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [A counterexample to metric-entropy duality](https://closedform.nishachay.tech/e/openai-math-2026-329/) - Functional analysis - [Independence of the separable quotient problem](https://closedform.nishachay.tech/e/openai-math-2026-323/) - Functional analysis - [The complete Crouzeix conjecture](https://closedform.nishachay.tech/e/openai-math-2026-325/) - Functional analysis - [Tingley’s sphere-isometry problem](https://closedform.nishachay.tech/e/openai-math-2026-322/) - Functional analysis - [The cotype–cotype conjecture under the approximation property](https://closedform.nishachay.tech/e/openai-math-2026-326/) - Functional analysis - [Functional analysis](https://closedform.nishachay.tech/science/mathematics/functional-analysis/) - Field · 11 results - [Open since the 1970s](https://closedform.nishachay.tech/open-since/1970s/) - Era · 63 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20324%5D%20)