The same metric shape can hide different linear Banach spaces

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Family 324 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A bounded linear isomorphism between Banach spaces preserves distances up to fixed multiplicative constants. Does the converse hold if a bijection preserves distances to that same accuracy but is allowed to be nonlinear?

Family 324 claims that the answer is no even for separable real Banach spaces. One manuscript distinguishes the spaces by whether they contain a linear copy of a sequence space with Hilbert-valued coordinates. The other gives a stronger ordinary-sequence obstruction: a space containing no linear copy of $c_0$ is nevertheless bi-Lipschitz equivalent to itself with an entire $c_0$ summand added.

These are global, whole-space equivalences with fixed two-sided distance bounds. They are stronger than earlier examples involving only uniform continuity or coarse control at large distances.

2. Background and essential definitions

A map $F:X\to Y$ is bi-Lipschitz if constants $0<c\leq C<\infty$ satisfy

$$c\|x-x'\|\leq\|F(x)-F(x')\|\leq C\|x-x'\|$$

for all pairs. It is a bi-Lipschitz equivalence if it is onto; the lower bound then makes it a bijection with Lipschitz inverse. A linear isomorphism additionally preserves addition and scalar multiplication and has bounded inverse.

A uniform homeomorphism requires both directions to be uniformly continuous, but need not provide linear bounds at every scale. A coarse-Lipschitz map can allow additive errors, which weaken its control of small distances. Neither automatically yields the theorem’s global inequalities.

The real Banach space $c_0$ consists of scalar sequences tending to zero, with supremum norm. More generally,

$$c_0(H)=\{(v_i):v_i\in H,\ \|v_i\|\to0\}, \qquad \|(v_i)\|=\sup_i\|v_i\|.$$

For $H=\ell^2$, each coordinate is a whole infinite-dimensional Hilbert space. Containing a linear copy of $c_0(\ell^2)$ is therefore a stronger structural condition than containing ordinary $c_0$.

The sum $X\oplus_\infty Y$ has norm $\max\{\|x\|,\|y\|\}$. A linear copy means a closed subspace linearly isomorphic to the indicated space, not merely a nonlinear image with comparable distances.

3. Development history and previous partial results

Ribe’s theorem shows that uniform homeomorphisms preserve finite-dimensional linear structure up to a uniform distortion bound. This is a powerful reason to expect metric geometry to constrain linear geometry, but it does not recover an entire infinite-dimensional linear isomorphism.

Aharoni–Lindenstrauss constructed nonseparable bi-Lipschitz equivalent Banach spaces that are not linearly isomorphic and asked for separable examples. Ribe later gave separable uniformly homeomorphic nonisomorphic spaces. Further work separated uniform equivalence from Lipschitz equivalence and expanded the collection of uniform counterexamples.

Positive rigidity results obstruct straightforward constructions. Godefroy–Kalton–Lancien proved that a space bi-Lipschitz equivalent to $c_0$ must be linearly isomorphic to $c_0$, and similarly preserved being isomorphic to a subspace of $c_0$. These concern whole-space equivalences, not merely containing a bi-Lipschitz image of $c_0$.

Godefroy–Kalton’s Lipschitz-free lifting theorem supplies another constraint: a Lipschitz right inverse of a bounded linear quotient onto a separable Banach space yields a bounded linear right inverse. A construction claiming to lift arbitrary quotient maps nonlinearly must therefore distinguish exactly what it is lifting.

Separable quasi-Banach counterexamples existed for exponents below one, but they do not settle the Banach-norm case. Kalton recorded the separable Lipschitz-isomorphism problem separately from the question whether a Lipschitz image of $c_0$ forces a linear copy.

M1 claims the former using a Hilbert-block obstruction. M2 addresses the ordinary $c_0$ embedding question and proves an onto absorption statement. Both identify earlier uniform or coarse constructions while retaining their weaker distance hypotheses.

4. What the new manuscripts claim

M1: separable spaces with different Hilbert-block subspaces

M1 constructs separable real Banach spaces $X,Y$ and an onto map $\Psi:X\to Y$ with

$$\frac4{21}\|s-t\|_X \leq\|\Psi(s)-\Psi(t)\|_Y \leq\frac{76}{25}\|s-t\|_X.$$

The space $X$ contains a linearly isometric copy of $c_0(\ell^2)$, while $Y$ contains no closed subspace linearly isomorphic to it. Therefore the spaces cannot be linearly isomorphic.

The obstruction does not say that $Y$ contains no ordinary $c_0$. M1 states neither reflexivity nor a complex-scalar analogue.

M2: metric absorption without a linear copy

M2 constructs a separable real Banach space $Z$ with no closed linear subspace isomorphic to $c_0$, together with an onto bi-Lipschitz map

$$F:Z\oplus_\infty c_0\longrightarrow Z.$$

Restricting to the added summand embeds $c_0$ bi-Lipschitzly into $Z$, even though no linear copy exists. The whole spaces are bi-Lipschitz equivalent but not linearly isomorphic, since the domain visibly has a linear $c_0$ subspace.

Aharoni’s theorem says every separable metric space embeds bi-Lipschitzly into $c_0$. Composing with the displayed embedding makes this same $Z$ a bi-Lipschitz universal target for separable metric spaces. This means existence of embeddings, not isometric universality or a claim that all those spaces become linearly embedded.

M2’s construction is independent of M1’s Hilbert-block and localized Lipschitz-free argument. Its source is dated September 26, while M1 is dated September 24.

5. Why the advance matters

Importance: 5/5. If correct, these examples settle a central rigidity question in nonlinear Banach-space geometry. Global distance equivalence would fail to determine linear isomorphism even under separability.

M2 sharpens the separation in a particularly recognizable way. An entire $c_0$ direction can be absorbed metrically by one space that cannot contain that direction linearly. Metric embeddings of all separable metric spaces can coexist with this linear exclusion.

The conclusions do not contradict exact-distance rigidity, such as the sphere-isometry theorem claimed in family 322. Fixed multiplicative distortion permits behavior that exact distance preservation forbids. They also do not contradict Ribe’s preservation of finite-dimensional structure; the distinguishing subspace properties are infinite-dimensional.

6. The main idea

Lift a correction, not a quotient inverse

Both constructions use a triangular change of variables. A nonlinear map rearranges coordinates, and an auxiliary space lifts the difference between that map and a simple projection. This is weaker than finding a Lipschitz right inverse for a linear quotient, so the Godefroy–Kalton splitting theorem does not directly force the unwanted linear structure.

In M2 the data are a bi-Lipschitz map $g:E\oplus_\infty U\to U$, a bounded linear map $Q:Z\to U$, and a Lipschitz correction $K$ satisfying

$$QK=g-P,$$

where $P$ projects onto $U$. They yield

$$F(b,t)=b+K(t,Qb).$$

The identity makes an explicit inverse possible. No surjectivity or right-inverse property is demanded of $Q$.

M1: local orthogonality becomes a linear obstruction

M1 first constructs a bi-Lipschitz change of Hilbert-space coordinates $h:U\oplus_2\ell^2\to U$. Its nonlinear correction has a local orthogonality property: near finitely many distinct input points, sufficiently high output coordinates occupy mutually orthogonal regions.

The Lipschitz-free space over the domain turns this correction into a bounded linear map $q$. A Lipschitz-free space is built from point evaluations so that zero-preserving Lipschitz maps admit linearizations. M1 proves that the local geometry makes $q$ completely continuous, sending weakly null sequences to norm-null sequences.

Sequence graph spaces combine a weighted summable norm with a supremum norm on their $q$-images. The nonlinear coordinate change acts on each sequence coordinate and gives the desired bi-Lipschitz equivalence. The domain visibly retains $c_0(\ell^2)$; complete continuity and a disjoint-support argument exclude it from the target.

M2: coordinate shifts with controlled nonlinear weights

For ordinary $c_0$, M2 builds $g$ using finite coordinate blocks arranged in rows. With weights frozen, the construction shifts scalar slots and has an explicit linear inverse. The actual weights depend on the input, so global invertibility requires controlling their variation, not just inverting the frozen map.

Finite-row arguments use Brouwer’s theorem, with an extra scalar balancing dimensions and controlling spillover. Uniform bounds and a closed-image argument then yield an onto map on the infinite space.

Design a norm that excludes linear c0

M2 records the coordinate corrections as localized scalar Lipschitz tests. It builds $Z$ from finite mass-zero combinations of points, using selected test lists and a norm combining exponents approaching one with an outer square sum.

If a linear $c_0$ basis existed, its finite signed sums would remain uniformly bounded. The test construction extracts one detecting function per basis vector while retaining uniform Lipschitz control on signed sums of those tests. Separated supports or carefully controlled nested supports then force the vector sums’ norms to grow, a contradiction.

The same tests define $Q$ and the lifted correction $K$, connecting the exclusion argument to the metric absorption map rather than constructing two unrelated spaces.

7. Scope, limitations, and open questions

Both manuscripts concern real separable Banach spaces and global bi-Lipschitz maps. They do not establish complex-linear counterexamples, reflexive examples, or a universal optimal distortion constant.

M1 excludes $c_0(\ell^2)$, not ordinary $c_0$; M2 supplies the latter exclusion by a different construction. Its universality statement concerns metric embeddings, not whole-space equivalences with every separable metric space.

The rigidity theorem for spaces bi-Lipschitz equivalent to $c_0$ remains compatible with M2. The target $Z$ contains a nonlinear image of $c_0$, but is not asserted bi-Lipschitz equivalent to $c_0$ itself. Likewise, the correction-lifting identity is not a nonlinear section of a linear quotient.

The examples leave many structured versions of nonlinear rigidity meaningful: imposing reflexivity, additional smoothness, or particular linear classes may change the answer. The present claims settle the unrestricted separable real questions stated above, not every such refinement.

8. Sources and further reading

• Godefroy, Kalton and Lancien, Subspaces of c0 and Lipschitz isomorphisms, for the positive whole-space rigidity results.
• Godefroy and Kalton, Lipschitz-free Banach spaces, for linearization and the quotient-section constraint.
• M1 and M2 state their distinct linear obstructions and explain why their correction lifts do not violate the established linear splitting theorem.

Source manuscripts

• M1. OpenAI, Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic (September 24, 2026).
• M2. OpenAI, Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0 (September 26, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.