Stability, hereditary containment, and the exponent quantifier

Definitions and mathematical statement

Problem 28 asks for a meaningful notion of an "explicitly defined norm" guaranteeing hereditary containment of an $\ell_p$ space. The excerpt does not define explicitness or specify whether $p$ is fixed for all subspaces. The supplied partial result uses Krivine--Maurey stability as a sufficient mathematical property and isomorphic containment, not an independently specified syntactic or computability definition of explicitness.

Fix a scalar field ${\mathbb{R}}$ or ${\mathbb{C}}$. A Banach space $X$ is stable in the stated sense if for all bounded sequences $(x_m)$, $(y_n)$ in $X$ and free ultrafilters $\mathcal U,\mathcal V$ on ${\mathbb{N}}$, \[ \lim_{m\to\mathcal U}\lim_{n\to\mathcal V}\|x_m+y_n\| =\lim_{n\to\mathcal V}\lim_{m\to\mathcal U}\|x_m+y_n\|. \] A copy of $\ell_p$, $1\le p<\infty$, in $Y$ means a bounded linear isomorphism $J:\ell_p\to J(\ell_p)\subseteq Y$ onto a closed subspace; its distortion is $\|J\|\|J^{-1}\|$. An almost-isometric copy means that for every $\varepsilon>0$ such an embedding can be chosen with distortion at most $1+\varepsilon$.

The aid invokes the stable-space theorem to assert the varying-exponent hereditary conclusion \[ \forall Y\subseteq X\text{ closed, infinite dimensional}\quad \exists p(Y)\in[1,\infty)\quad \forall\varepsilon>0\ \exists J:\ell_{p(Y)}\to Y, \quad\|J\|\|J^{-1}\|\le1+\varepsilon, \] whenever $X$ is stable. The exponent is chosen for $Y$ and works for every approximation tolerance; it need not be independent of $Y$.

The stronger quantifier order \[ \exists p\in[1,\infty)\quad \forall Y\subseteq X\text{ closed, infinite dimensional},\quad Y\text{ contains an isomorphic copy of }\ell_p \] is not asserted for all stable spaces. The supplied obstruction is \[ X=\ell_1\oplus_2\ell_2,\qquad \|(u,v)\|=(\|u\|_1^2+\|v\|_2^2)^{1/2}. \] The aid reports this space stable, but its two coordinate subspaces admit no common $\ell_p$ exponent: the Schur property of $\ell_1$ excludes $\ell_p$ for $p>1$, whereas reflexivity of $\ell_2$ excludes $\ell_1$.

A Banach space $X$ is minimal if it is isomorphic to a subspace of each of its infinite-dimensional closed subspaces. For $C\ge1$, call it $C$-minimal here if each such $Y$ admits an embedding $I_Y:X\to Y$ with distortion at most $C$. Under this explicit convention the aid asserts \[ X\text{ stable and }C\text{-minimal}\quad\Longrightarrow\quad \exists p\in[1,\infty)\ \forall Y\subseteq X\text{ closed, infinite dimensional}\ \forall\varepsilon>0\ \exists J_Y:\ell_p\to Y, \quad\|J_Y\|\|J_Y^{-1}\|\le C(1+\varepsilon). \] For minimal stable spaces without a common distortion constant, the same fixed-exponent isomorphic-containment conclusion holds, with no uniform distortion bound asserted. These are additional-hypothesis statements, not a definition-independent answer about all explicitly described norms.

The accompanying Definable Norms and Stable Banach Spaces working-group note belongs to Problem 28. The trailing "29. Study asymptotic cones" and its Asymptotic Cones working-group note are a separate problem, according to the supplied source-verification report.

Short English statement

Under the aid's stability interpretation, every infinite-dimensional closed subspace contains some almost-isometric $\ell_p$, with $p$ allowed to vary. A fixed exponent fails for general stable spaces but works under minimality, with bound $C(1+\varepsilon)$ under the stated $C$-minimality convention.

Sources

• Local registry, permanent UnsolvedMath ID $20002337$, exact public code catalogue entry; canonical location aim-logic-notes.json, notes[112].
• AIM, Questions on Model theory for metric structures, Problem 28 and Definable Norms and Stable Banach Spaces working-group background. Supplied reference: https://aimath.org/WWN/continuouslogic/continuouslogic.pdf. The verification field reports that the official document separates Problem 29 from the concatenated registry extraction; this is supplied history, not an inspection performed here.
• Krivine--Maurey stability and the stable Banach space containment theorem are invoked by the supplied aid without a full bibliographic citation. No proposer, year, original remarks, or literature entry is supplied.
• Supplied research attempt 1: partial_result, run and registry status partially_solved, rigor label proved, low novelty confidence; classification PARTIAL-PROGRESS, created and updated 14 August 2026. The aid is machine generated, not peer reviewed, and requires independent verification before citation.