Banach Spaces and Ramsey Theory — Problem 7

Definitions and mathematical statement

A Banach space is separable if it has a countable dense subset. Two normed spaces over the same scalar field are isomorphic, written $Y\cong Z$, if there is a bounded linear bijection between them with bounded inverse. A Banach space is Hilbertian if it is isomorphic to a Hilbert space; non-Hilbertian means that no such isomorphism exists.

Source question (G. Godefroy, reference label [Go]). Let $X$ be a separable non-Hilbertian Banach space. Does there exist a sequence $(X_n)_{n\geq 1}$ of subspaces of $X$, endowed with their inherited norms, such that \[ X_m\not\cong X_n\qquad\text{for every }m,n\geq 1\text{ with }m\neq n? \] The supplied source does not specify whether the scalar field is ${\mathbb{R}}$ or ${\mathbb{C}}$, whether the subspaces must be closed, or whether they must be infinite-dimensional. These qualifications remain unresolved. In particular, the intended admissibility of finite-dimensional subspaces must be clarified before this can be treated as a precise substantive open problem. Under the additional interpretation that all subspaces are closed and infinite-dimensional, the requested sequence would satisfy \[ \forall n\geq 1,\quad X_n\subseteq X\text{ is a closed linear subspace and }\dim X_n=\infty, \qquad \forall m\neq n,\quad X_m\not\cong X_n. \] This additional interpretation is a conditional formulation, not an explicit hypothesis in the supplied excerpt.

The source introduces the question by reporting that every separable non-Hilbertian Banach space contains two non-isomorphic subspaces. That contextual assertion is retained as supplied background, with the same unresolved subspace conventions.

Short English statement

Godefroy asks whether every separable Banach space that is not isomorphic to a Hilbert space contains a countably infinite family of pairwise non-isomorphic subspaces. The source leaves the scalar field, closedness, and infinite-dimensionality requirements unstated; the intended subspace convention requires review.

Sources

• Local UnsolvedMath registry: permanent ID $800744$, source problem code catalogue entry, title Banach Spaces and Ramsey Theory --- Problem 7. The registry supplies the status open; this label is not an independently verified current mathematical assessment.
• Dodos and Lopez-Abad, Banach Spaces and Ramsey Theory: Some Open Problems (2010), Problem 7. Supplied source link: https://arxiv.org/abs/1006.2668. The record identifies the extraction method as arxiv_latex and gives an access date of 2026-10-16. These are supplied provenance statements, not a claim of independent source access or verification.
• The problem is explicitly attributed in the supplied statement to G. Godefroy, with reference label [Go]. The bibliographic entry corresponding to [Go] is not included in the supplied record. The registry lists Dodos and Lopez-Abad as proposers and 2010 as the proposed year.