Lipschitz equivalent Banach spaces need not be linearly isomorphic

1 · The essentials

A Banach space is a setting where vectors can be added, scaled, and measured, and where converging approximations have limits inside the space. The question here is whether matching two such spaces by distances also forces their vector structures to match. The distance match allows stretching and shrinking, but only within fixed bounds that apply everywhere. The main manuscript gives two spaces that match in this sense but cannot be matched by a reversible, continuous map that respects addition and scaling. Both spaces are real and separable, meaning that each has a countable collection of points that can approximate every point. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two elastic grids whose points can be paired without stretching or shrinking any distance beyond fixed limits. That pairing need not respect the grids’ rules for adding arrows: the result separates a controlled distance match from a match of vector operations.

Why it matters

The result distinguishes two ways of classifying Banach spaces: by controlled changes of distance and by their linear structure. Even a global bi-Lipschitz match does not determine the linear isomorphism class of a separable real Banach space. The companion construction makes this distinction concrete by absorbing an extra space in the distance-based sense without containing a linear copy of it.

2 · Going further

The precise question is whether globally bi-Lipschitz equivalent separable real Banach spaces must be linearly isomorphic. A bi-Lipschitz equivalence is a bijection $f : X \to Y$ for which constants $0 < a \leq b < \infty$ satisfy $a ∥ x - x^{'} ∥ \leq ∥ f (x) - f (x^{'}) ∥ \leq b ∥ x - x^{'} ∥$ for all $x, x^{'} \in X$ . A linear isomorphism, by contrast, is a bounded linear bijection with bounded inverse. The main manuscript gives a negative answer to this Lipschitz-isomorphism problem.

The companion manuscript supplies the concrete construction described in the abstracts: a separable real Banach space $Z$ containing no linear copy of $c_{0}$ , yet bi-Lipschitz equivalent to $Z \oplus_{\infty} c_{0}$ . Here $c_{0}$ is the space of real sequences tending to zero, with the supremum norm, and the direct sum has norm $∥ (z, u) ∥ = max {∥ z ∥, ∥ u ∥}$ . Thus adding an entire $c_{0}$ component leaves the space unchanged up to bi-Lipschitz equivalence.

This construction explains the obstruction to a linear match: $Z \oplus_{\infty} c_{0}$ contains its explicit $c_{0}$ component, whereas $Z$ contains no linear copy of $c_{0}$ . A linear isomorphism would carry that component to a linear copy inside $Z$ . The companion abstract also states that the same $Z$ contains a bi-Lipschitz image of every separable metric space. The abstracts do not describe how $Z$ or the distance-preserving-with-bounds maps are constructed; both manuscripts are labeled Lean-formalized in the supplied source.

Useful notions

Norms and distances
Complete normed vector spaces
Countable dense subsets
Linear maps and bounded inverses
Bi-Lipschitz maps

What was known before / what is new

Before

The source identifies the separable Banach-space Lipschitz-isomorphism problem as the question being resolved. It supplies no account of earlier partial results or the history of the problem.

New

The main result establishes the existence of globally bi-Lipschitz equivalent separable real Banach spaces that are not linearly isomorphic. The companion manuscript constructs a space $Z$ that absorbs $c_{0}$ bi-Lipschitzly while containing no linear copy of $c_{0}$ . It also states that this same space contains a bi-Lipschitz image of every separable metric space.

Limits and assumptions

The counterexample concerns separable real Banach spaces and a global bi-Lipschitz equivalence; it does not assert an exact preservation of distances. It is an existence result, not a claim that every bi-Lipschitz equivalent pair fails to be linearly isomorphic. The main result has a Lean formalization, and the companion manuscript is also labeled Lean-formalized, but the supplied abstracts do not specify the claim-by-claim scope of formal verification. They provide neither construction details nor further open questions.

Mini glossary

Banach space: A normed vector space in which every Cauchy sequence converges to a point of the space.
Separable: Having a countable dense subset, whose points approximate every point of the space.
Bi-Lipschitz equivalence: A bijection that bounds every distance in its image above and below by fixed positive multiples of the original distance.
Linear isomorphism: A bounded linear bijection between Banach spaces with a bounded inverse.
$c_{0}$: The Banach space of real sequences tending to zero, equipped with the supremum norm.
Linear copy: A linear subspace that is linearly isomorphic, with its inherited norm, to the specified Banach space.

Keywords : Banach spaces · Bi-Lipschitz equivalence · Linear isomorphism

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/324.md

Manuscripts in this family

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Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

24/09/2026

Verified by 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 GitHub folder Lean formalization

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

26/09/2026

Verified by Lean

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.

PDF GitHub folder Lean formalization