A ball between c₀₀ and ℓ∞ that is the nonexpansive bijective image of a ball between c₀₀ and c₀ lies in c₀

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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 $\|x\|=\sup_{n\in\Gamma}|x_n|$, let $c_0(\Gamma)$ be the subspace of $x$ for which $\{n:|x_n|\ge\varepsilon\}$ is finite for every $\varepsilon>0$, and let $c_{00}(\Gamma)$ be the subspace of finitely supported $x$; likewise for $\Delta$. For a linear subspace $X\subseteq\ell_\infty(\Gamma)$, not assumed closed, write $B_X=\{x\in X:\|x\|\le1\}$.

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 $c_{00}(\Delta)\subseteq Y$, with no upper bound assumed.
3. $F:\ell_\infty(\Gamma)\to\ell_\infty(\Delta)$ is a map whose values outside $B_X$ play no role.
4. $F$ maps $B_X$ into $B_Y$, is injective on $B_X$, and every point of $B_Y$ equals $F(x)$ for some $x\in B_X$.
5. $\|F(x)-F(x')\|\le\|x-x'\|$ for all $x,x'\in B_X$.

There is no sign-pattern or "$X=c_{00}$" hypothesis on this card.

The conclusion is $Y\subseteq c_0(\Delta)$: every $y\in Y$ tends to $0$ along the cofinite filter of $\Delta$. Together with hypothesis 2 this places $Y$ between $c_{00}(\Delta)$ and $c_0(\Delta)$.

Flags

Lean conventions that matter. The conventions are those listed under the first card: sup norm, cofinite-filter $c_0$, non-closed subspaces allowed, BijOn on the ball only, LipschitzOnWith 1 as nonexpansiveness on the ball. The conclusion Y ≤ c0 Δ is inclusion of submodules; since $Y=\mathbb R\cdot B_Y$, it is equivalent to $B_Y\subseteq c_0(\Delta)$.

Degenerate cases.

• $\Delta$ finite: the conclusion is trivially true ($c_0(\Delta)=\ell_\infty(\Delta)$), whatever the hypotheses. The card has content only for infinite $\Delta$.
• $\Gamma$ empty forces $\Delta$ empty, and $\Gamma$ finite forces $\Delta$ finite (compactness, as under the first card); the conclusion holds in both cases.
• $\Gamma$ infinite and $\Delta$ finite: the conclusion holds trivially (the hypotheses are in fact unsatisfiable by invariance of domain, but this does not matter).

Hypotheses, one at a time.

• hX0 ($c_{00}(\Gamma)\subseteq X$): undecided; possibly removable, but not by an easy argument. Without it $X$ is an arbitrary subspace of $c_0(\Gamma)$. When $X$ is finite-dimensional (including $X=\{0\}$), $B_X$ is compact, so $B_Y$ is compact, so $\Delta$ is finite and the conclusion holds. When $F$ is an isometry, $Y$ is linearly isometric to $X\subseteq c_0(\Gamma)$, hence its completion is polyhedral, while $Y\not\subseteq c_0(\Delta)$ would put an isometric copy of $c$ into the closure of $Y$; so the conclusion holds. For non-isometric $F$ and infinite-dimensional $X\not\supseteq c_{00}(\Gamma)$ I cannot decide.
• hX1 ($X\subseteq c_0(\Gamma)$): needed. Counterexample without it: $X=Y=\ell_\infty(\mathbb N)$ and $F$ the identity.
• hY0 ($c_{00}(\Delta)\subseteq Y$): needed. Counterexample without it: $\Gamma=\mathbb N$, $\Delta=\mathbb N\sqcup\mathbb N'$ (two copies), $X=c_0(\mathbb N)$, and $T x$ equal to $x$ on the first copy and to the constant $x_0$ on the second copy. $T$ is a linear isometry, so $F=T$ is a nonexpansive bijection of $B_X$ onto $B_{T(X)}$, and $Y=T(X)$ contains $Te_0\notin c_0(\Delta)$.
• hbij: needed, since without it $Y=\ell_\infty(\mathbb N)$ is unconstrained (take $F=0$). I did not settle whether its injectivity half alone could be dropped; the surjectivity half is plainly essential.
• hlip: needed. Counterexample without it: $X=c_0(\mathbb N)$, $Y=\ell_\infty(\mathbb N)$; both balls have the cardinality of the continuum, so a set-theoretic bijection exists, and $Y\not\subseteq c_0$.

Name. le_c0_of_bijOn_ballIn correctly says that the target subspace lies in $c_0$ given a bijection of the balls. It omits the nonexpansive hypothesis, which is essential (see hlip); a reader of the name alone might think bijectivity suffices. This is a mild naming issue; something like le_c0_of_nonexpansive_bijOn_ballIn would be more accurate.

Truth as written. The card is implied by the third card (exists_signed_permutation_inward) and, under the extra hypothesis 3 of the first card, by the first card. Its degenerate and finite cases hold. The infinite case is research-level and I cannot verify it from the statement; I found no counterexample. This is the card where the wide range of $Y$ (between $c_{00}(\Delta)$ and $\ell_\infty(\Delta)$) is the whole content. The obstructions listed under the first card all point the same way: density forces $|\Delta|\le|\Gamma|$ for infinite $\Gamma$, isometric counterexamples are impossible by the polyhedrality argument, and for $X=c_{00}(\mathbb N)$ the target ball must be $\sigma$-compact. A quick test also fits: for $X=c_0(\mathbb N\sqcup\{*\})$ and $Y=c$ (convergent sequences), no map of the form $F(x)_k=\varphi(x_*,x_k)$ with $\varphi$ nonexpansive reaches all of $B_c$. Reaching $\mathbf 1-2e_1$ would need limit $\varphi(s,0)=1$ and $\varphi(s,a)=-1$ with $|a|\le1$, contradicting $|\varphi(s,a)-\varphi(s,0)|\le|a|$. I believe the card is true, with the caveat that I have not seen or checked a proof.