Nonexpansive bijections from balls between c₀₀ and c₀ onto balls between c₀₀ and ℓ∞ are signed permutations, wherever the target matches signs

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Let $\Gamma$ and $\Delta$ be arbitrary index sets (types), possibly empty, finite or uncountable, and possibly of different cardinalities. Let $\ell_\infty(\Gamma)$ be the real vector space of bounded functions $x:\Gamma\to\mathbb R$ with the supremum norm $\|x\|=\sup_{n\in\Gamma}|x_n|$ (the zero space when $\Gamma=\varnothing$). Inside it, $c_0(\Gamma)$ is the subspace of those $x$ that tend to $0$ along the cofinite filter, that is, $\{n\in\Gamma:|x_n|\ge\varepsilon\}$ is finite for every $\varepsilon>0$, and $c_{00}(\Gamma)$ is the subspace of finitely supported $x$. For a linear subspace $X\subseteq\ell_\infty(\Gamma)$, not assumed closed, write

$$B_X=\{x\in X:\ \|x\|\le 1\}\subseteq\ell_\infty(\Gamma).$$

Say that a subspace $Y\subseteq\ell_\infty(\Delta)$ realizes sign patterns if for every countable set $C\subseteq\Delta$ and every function $\varepsilon:\Delta\to\mathbb R$ with $\varepsilon_d\in\{1,-1\}$ for all $d\in C$ there is $v\in Y$, of any norm, with $\varepsilon_d\,v_d>0$ for every $d\in C$.

The hypotheses are:

1. $X$ is a real linear subspace of $\ell_\infty(\Gamma)$ with $c_{00}(\Gamma)\subseteq X\subseteq c_0(\Gamma)$.
2. $Y$ is a real linear subspace of $\ell_\infty(\Delta)$ with $c_{00}(\Delta)\subseteq Y$. No upper bound on $Y$ is assumed, so $Y=\ell_\infty(\Delta)$ is allowed.
3. Either $X=c_{00}(\Gamma)$, or $Y$ realizes sign patterns.
4. $F:\ell_\infty(\Gamma)\to\ell_\infty(\Delta)$ is a map whose values outside $B_X$ play no role.
5. $F$ maps $B_X$ into $B_Y$, is injective on $B_X$, and every point of $B_Y$ equals $F(x)$ for some $x\in B_X$.
6. $\|F(x)-F(x')\|\le\|x-x'\|$ for all $x,x'\in B_X$.

The conclusion is that there exist a bijection $\sigma:\Gamma\to\Delta$ and a function $\alpha:\Gamma\to\mathbb R$ with $\alpha_n\in\{1,-1\}$ for every $n\in\Gamma$ such that

$$F(x)_{\sigma(n)}=\alpha_n\,x_n\qquad\text{for every }x\in B_X\text{ and every }n\in\Gamma.$$

Since $\sigma$ is onto, this determines every coordinate of $F(x)$: $F(x)_d=\alpha_{\sigma^{-1}(d)}\,x_{\sigma^{-1}(d)}$ for all $d\in\Delta$. In particular the conclusion asserts that $\Gamma$ and $\Delta$ have the same cardinality.

Flags

Lean conventions that matter.

• Linf Γ is Mathlib's lp (fun _ => ℝ) ⊤, the real $\ell_\infty(\Gamma)$ with the sup norm; for empty $\Gamma$ it is the zero space.
• c0 Γ is defined through the cofinite filter. For finite $\Gamma$ it is all of $\ell_\infty(\Gamma)$; for uncountable $\Gamma$ its elements have countable support.
• c00 Γ is finite support; for finite $\Gamma$ one has $c_{00}(\Gamma)=c_0(\Gamma)=\ell_\infty(\Gamma)$.
• ballIn X is $X$ intersected with the closed unit ball of the ambient $\ell_\infty$. Neither $X$ nor $Y$ needs to be closed, so dense subspaces such as $c_{00}$ are in scope.
• Set.BijOn F (ballIn X) (ballIn Y) means maps-into, injective on $B_X$, and onto $B_Y$; it says nothing about $F$ off $B_X$. This is exactly "the restriction of $F$ is a bijection of $B_X$ onto $B_Y$".
• LipschitzOnWith 1 F (ballIn X) uses the extended distance of the sup norm, so it is nonexpansiveness on $B_X$. No regularity of the inverse is assumed.
• RealizesSignPatterns quantifies only over countable $C$, leaves $\varepsilon$ arbitrary off $C$, and puts no norm bound on $v$. It holds for every $Y\supseteq c_0(\Delta)$ (enumerate $C=\{d_1,d_2,\dots\}$ and take $v=\sum_k 2^{-k}\varepsilon_{d_k}e_{d_k}$), for every admissible $Y$ when $\Delta$ is finite, and it fails for $Y=c_{00}(\Delta)$ when $\Delta$ is infinite. It is monotone in $Y$.
• Hypothesis 3 is automatic when $\Gamma$ is finite (then $X=c_{00}(\Gamma)$), when $\Delta$ is finite, or when $Y\supseteq c_0(\Delta)$. It excludes only: $\Gamma$ infinite, $X\ne c_{00}(\Gamma)$, and $Y$ missing some sign pattern on a countably infinite set, for example $X=c_0(\mathbb N)$ with $Y=c_{00}(\mathbb N)$.
• Hypothesis 3 is asymmetric: its first disjunct is about $X$, its second about $Y$. Under the conclusion, $Y$ is the image of $X$ under a signed relabelling of coordinates, so $Y$ realizes sign patterns exactly when $X$ does; if the card is true, hypothesis 3 can be read as a condition on $X$ alone.
• $\sigma$ has type Γ ≃ Δ, an equivalence between possibly different types (even in different universes).

Degenerate cases.

• $\Gamma=\Delta=\varnothing$: the conclusion holds with the empty bijection.
• Exactly one of $\Gamma,\Delta$ empty: the hypotheses cannot hold, because one ball is $\{0\}$ and the other contains $0$ and some $e_d$; the card is vacuous there.
• $\Gamma$ a single point: $B_X=[-1,1]$, and $B_Y$ is a continuous injective image of an interval, hence an arc. If $|\Delta|\ge2$ then $B_Y$ contains a square, so $\Delta$ is a single point and $Y=\mathbb R$. A nonexpansive bijection of $[-1,1]$ onto itself is $\pm$ the identity, so the conclusion holds.
• $\Gamma$ finite: $B_X$ is compact, so $B_Y$ is compact; the vectors $e_d$ are $1$-separated in $B_Y$, so $\Delta$ is finite and $Y=\ell_\infty(\Delta)$. Invariance of domain gives $|\Gamma|=|\Delta|$, finite-dimensional plasticity makes $F$ an isometry, Mankiewicz's theorem and central symmetry make it linear, and linear isometries of $\ell_\infty^n$ are signed permutations. The conclusion holds.

Hypotheses, one at a time.

• hX0 ($c_{00}(\Gamma)\subseteq X$): needed. Counterexample without it: $\Gamma=\{1\}$, $\Delta=\varnothing$, $X=Y=\{0\}$, $F=0$. Hypothesis 3 holds vacuously for $\Delta=\varnothing$, $F$ is a nonexpansive bijection of $\{0\}$ onto $\{0\}$, and there is no bijection $\{1\}\to\varnothing$.
• hX1 ($X\subseteq c_0(\Gamma)$): not removable on present knowledge. Without it, the instance $X=Y=\ell_\infty(\mathbb N)$ (which satisfies hypothesis 3) would assert that every nonexpansive bijection of the $\ell_\infty(\mathbb N)$ ball onto itself is a signed permutation, a plasticity question I do not know to be settled. I know no counterexample.
• hY0 ($c_{00}(\Delta)\subseteq Y$): needed. Counterexample without it: $\Gamma=\{1\}$, $\Delta=\{1,2\}$, $X=\mathbb R=c_{00}(\Gamma)$, $Y=\mathbb R e_1$, $F(x)=x_1e_1$. $F$ is an isometric bijection of the balls, the first disjunct of hypothesis 3 holds, and there is no bijection from a one-point set onto a two-point set.
• hmatch (hypothesis 3): not implied by the others, since $X=c_0(\mathbb N)$, $Y=c_{00}(\mathbb N)$ meet hX0, hX1, hY0 and violate it. Whether the statement survives without it I cannot decide; the uncovered case contains, for example, the question whether a nonexpansive bijection of $B_{c_0(\mathbb N)}$ onto $B_{c_{00}(\mathbb N)}$ exists. I found no counterexample.
• hbij: needed. With $\Gamma=\Delta=\{1\}$ and $X=Y=\mathbb R$, the nonexpansive map $F=0$ violates the conclusion. Each half matters: without injectivity, $\Gamma=\{1,2\}$, $\Delta=\{1\}$, $F(x)=x_1$ is a nonexpansive surjection of the square onto $[-1,1]$ and no bijection $\Gamma\to\Delta$ exists; without surjectivity, $F(x)=x/2$ on $\Gamma=\Delta=\{1\}$ fails. I did not test dropping only the maps-into part.
• hlip: needed. With $\Gamma=\Delta=\{1\}$, $X=Y=\mathbb R$, let $F$ swap $\tfrac12$ and $-\tfrac12$ and fix every other point of $[-1,1]$; then $\alpha=1$ fails at $x=\tfrac12$ and $\alpha=-1$ fails at $x=1$.

Name. exists_signed_permutation_eq describes the conclusion: a signed bijection of index sets that equals $F$ on the ball. "Permutation" here means a bijection $\Gamma\to\Delta$ between two possibly different sets, which the type makes clear. The name is silent on hypothesis 3; I see no naming problem.

Truth as written. I checked the degenerate and finite-dimensional cases above and they hold. The infinite-dimensional case is a research-level statement that I cannot verify from the statement alone; I found no counterexample. On the target $Y$ being allowed up to $\ell_\infty(\Delta)$: the conclusion forces $Y$ to be the signed relabelling of $X$, hence $Y\subseteq c_0(\Delta)$, so under hypothesis 3 this card contains the statement of le_c0_of_bijOn_ballIn. I found no way the wider $Y$ makes the card false, and three cheap obstructions point the same way. First, a $1$-Lipschitz surjection cannot raise density, so $|\Delta|\le|\Gamma|$ for infinite $\Gamma$, and for instance $Y=\ell_\infty(\mathbb N)$ with countable $\Gamma$ is impossible. Second, no isometric $F$ can be a counterexample: an isometry of balls is linear (Mankiewicz plus central symmetry), if $Y\not\subseteq c_0(\Delta)$ then the closure of $Y$ contains an isometric copy of $c$ (clip some $y\in Y\setminus c_0(\Delta)$ at its limsup and correct it by a $c_0$ vector along a sequence realizing that limsup), $c$ is not polyhedral while every subspace of $c_0(\Gamma)$ is, and once $\overline{Y}=c_0(\Delta)$ a linear isometry $c_0(\Gamma)\to c_0(\Delta)$ is a signed bijection. So any counterexample would need a non-isometric nonexpansive bijection. Third, for $X=c_{00}(\mathbb N)$ the ball $B_X$ is $\sigma$-compact, so $B_Y$ must be too, which rules out, for example, $Y=c_0(\mathbb N)$. One point a human may want to confirm in the proof: the conclusion includes $|\Gamma|=|\Delta|$ also for uncountable $\Gamma$ and countable $\Delta$, and separability alone gives no obstruction there (the ball of $c_0(\omega_1)$ admits a continuous injection into a separable metric space), so the proof must get this from nonexpansiveness and bijectivity, not from a density count.