Tingley's problem
OpenAI's solution (2026) of Tingley's problem: every surjective isometry between the unit spheres of real normed spaces extends to a surjective real-linear isometry of the whole spaces. The course proves the Mazur–Ulam theorem, makes a hypothetical positive defect attained in an ultrapower, finds aligned extremal chords as a fixed point of a return map with Darbo's theorem (proved from Schauder's and Brouwer's), and rules them out by support, secant and convexity estimates.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Supports, radial contact and the Mazur–Ulam theorem
- Ultrapowers and Darbo's fixed-point theorem
- Aligned chords
- Tingley's problem
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
- Lebesgue's covering theorem and the dimension of cubes · Index theory of elliptic operators (Index theory of elliptic operators — Hermite and Schwartz tools)
- Convex functions of one real variable, the intermediate value theorem and compactness in Rn (core course Real Analysis I, C10).