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Tingley's problem: does every surjective isometry between unit spheres of Banach spaces extend to a linear isometry?

Theorem 1.1: if $X,Y$ are nonzero real Banach spaces and $f:S_X\to S_Y$ is a surjective isometry, then $T(0)=0$ , $T(x)=\|x\|f(x/\|x\|)$ is a surjective real-linear isometry, the unique linear extension of $f$ . No separability, reflexivity, smoothness, strict convexity or finite-dimensionality is assumed. For complex spaces only real linearity is concluded. The proof uses a maximal-defect argument, common supports and Darbo's fixed-point theorem. Incomplete normed spaces are not treated.

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Tingley's problem: does every surjective isometry between unit spheres of Banach spaces extend to a linear isometry? · Tingley's problem

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VibeMathed record: Tingley's problem: does every surjective isometry between unit spheres of Banach spaces extend to a linear isometry?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Tingley's question as the paper states it from Tingley (1987); it claims that for nonzero real Banach spaces the radial map $T(x)=\|x\|f(x/\|x\|)$ is a surjective real-linear isometry and the unique linear extension. The proof was not refereed. lean/formalization.yaml lists a main result (comparator TingleySphereIsometry, declaration OAI.Tingley.tingley_sphere_isometry_full, file OAI/Analysis/SphereIsometry/Extension.lean). ComparatorChallenges/TingleySphereIsometry.lean was read here: for complete, nontrivial real normed spaces of any size and a surjective isometry between unit spheres, it asserts a linear isometric equivalence agreeing with $f$ on the sphere, given by the radial formula, and unique among real-linear maps extending $f$ . This states the headline claim. Not rebuilt here.