What the note proves and explains
For each ε>0, the note constructs separable real Banach spaces Xε and Yε and a bijection Ψε fixing zero, with both Ψε and its inverse having Lipschitz constant at most 1+ε, while the spaces are not linearly isomorphic. The pair of spaces may depend on ε. The linear obstruction distinguishes a copy of c0(ℓ2) in Xε from its absence in Yε; it does not assert that Yε contains no ordinary c0.
The counterexample and its graph-space reduction are due to OpenAI. The original Theorem 1.1 has fixed numerical distance bounds. The arbitrary-ε estimates here are proved through a parameter-dependent reconstruction of that mechanism, not quoted as the literal statement of the original theorem. This is an exposition, not a claim of independent solution or priority.
The Hilbert-space construction is developed through an output-dependent inverse map. Finite-coordinate directions, separated transition intervals, a finite telescoping sum, and direct two-point estimates lead to a coordinate-by-coordinate proof of surjectivity. The figure and intermediate calculations are included to make this part of the argument readable without reconstructing omitted computations.
Intended readers
The note is for readers familiar with Banach and Hilbert spaces, bounded linear operators, weak convergence, and basic Lipschitz geometry. The Lipschitz-free-space inputs are stated where they are needed. Their detailed proofs can be read in the published companion, Lipschitz-Free Spaces over Hilbert Spaces, Version 1.
Reading route
Begin with Section 1 for the precise statement and the reduction to a Hilbert-space map. Section 2 identifies the free-space tools and their companion references. Section 3 constructs the weighted graph spaces and separates the metric equivalence from the linear obstruction. Section 4 proves the local-orthogonality criterion for complete continuity. Section 5 supplies the Hilbert-space map, including the finite-sum estimates and coordinate recursion. Section 6 chooses the parameters and assembles the final pair of spaces.
Materials
Read the complete Expository Note PDF, with linked contents, detailed proofs, the transition-interval figure, and references. The published companion PDF provides the background free-space arguments.
Original source. OpenAI, Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, preprint dated 24 September 2026. Read the original preprint or browse its source directory in OpenAI’s math repository. Both links identify the fixed manuscript used for this note.
Publication details
Expository Note · Version 1 · 49 pages.
Prepared 27 October 2026; published 28 October 2026.
Publication responsibility: Ryotaro Tanaka.
Suggested reference. Reading OpenAI's Construction of Lipschitz Homeomorphic but Linearly Nonisomorphic Separable Banach Spaces. Exact Mathematics with AI — Expository Note, Version 1. Published 28 October 2026; prepared 27 October 2026. Publication responsibility: Ryotaro Tanaka. PDF.
Responsibility and corrections
Publication responsibility: Ryotaro Tanaka. Publication responsibility covers release, versioning and corrections; it does not assert historical priority for this proof arrangement. The manuscript was drafted and revised using ChatGPT. Corrections should identify the statement or formula concerned and the relevant version; any subsequent correction will be recorded with that version.
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