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Math-2706 · mathematics · Analysis

Tingley extension for real Banach spaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any nonzero real Banach spaces X and Y, must every surjective isometry from the unit sphere of X onto the unit sphere of Y extend to a unique surjective real-linear isometry $X->Y$?

Short English statement

For any nonzero real Banach spaces X and Y, must every surjective isometry from the unit sphere of X onto the unit sphere of Y extend to a unique surjective real-linear isometry $X->Y$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2706-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2706

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 322; 2026-09-23).

Tingley extension for real Banach spaces

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 322; 2026-09-23), the asserted result is that an onto distance-preserving map between two real Banach unit spheres is the restriction of exactly one onto linear isometry.

Mathematical statement of the claimed result

Manuscript 1.
Whenever $X,Y$ are nonzero real Banach spaces and $f:S_X\to S_Y$ is a surjective distance-preserving map between unit spheres, the claimed extension is \[ T(0)=0,\qquad T(x)=\|x\|f(x/\|x\|)\quad(x\ne0). \] It is asserted to be a surjective real-linear isometry, and the only linear map extending $f$. No finite-dimensional or smoothness restriction is imposed.

Outline of the argument

For manuscript 1 (family 322; 2026-09-23), the radial extension's maximal cross-radius distance defect is attained after enlarging the spaces if necessary. Common supports propagate extremality to convex direction sets, and a Darbo fixed point aligns chords whose support and convexity estimates contradict a positive defect.

Formal verification

For manuscript 1 (family 322; 2026-09-23), OpenAI lists Lean statements for this assertion: an onto distance-preserving map between two real Banach unit spheres is the restriction of exactly one onto linear isometry.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are TingleySphereIsometry.lean.

Their named declarations include OAI.Tingley.tingley_sphere_isometry_full.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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