Results on supports, ultrapowers, fixed points, aligned chords and Tingley's problem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Common supports — Lemma 1.1. If ||alpha v + beta w|| = alpha + beta for unit v, w and alpha, beta > 0, then v and w share a support; a functional of norm at most 1 equal to 1 at a positive combination of unit vectors equals 1 on their hull.
• Radial contact — Lemma 2.1. For ||c|| < 1 each ray c + R_+ v, v in S_X, meets the sphere in exactly one point c + rho(v) v, with 1 - ||c|| <= rho <= 1 + ||c|| and rho continuous.
• The Mazur–Ulam theorem — Theorem 3.1. A surjective isometry between real normed spaces is affine.
• Reduction to the defect — Proposition 4.2. If a surjective sphere isometry has defect 0, its radial extension is a surjective real-linear isometry, the only linear extension.
• Limits along an ultrafilter — Lemma 1.2. Bounded real sequences have unique limits along a free ultrafilter on N; these limits are linear, monotone, commute with continuous functions and extend ordinary limits.
• Attainment of the defect — Proposition 3.1. A surjective sphere isometry of positive defect M between normed spaces yields one between Banach spaces (completed ultrapowers) with defect bound M attained at some (x_0, y, t), 0 < t < 1.
• Kuratowski's measure of noncompactness — Lemma 4.1. chi is unchanged by closure and by (closed) convex hulls; nested closed sets with chi -> 0 in a Banach space have nonempty compact intersection.
• Schauder's fixed-point theorem — Theorem 5.1. A continuous self-map of a nonempty compact convex subset of a normed space has a fixed point (from Brouwer's theorem).
• Darbo's fixed-point theorem — Theorem 6.1. A continuous self-map of a nonempty closed bounded convex subset of a Banach space with chi(g(H)) <= gamma chi(H), gamma < 1, has a fixed point.
• Propagation along common supports — Lemma 2.3. If v is extremal and l shares a support with the return direction g(v), then l is extremal; g preserves the extremal set.
• An invariant convex set — Lemma 3.1. There is a nonempty closed convex set of extremal directions preserved by the return map.
• Contraction of noncompactness — Lemma 4.2. On the invariant set, chi(g(H)) <= (1 - M/(1+t))^2 chi(H).
• Aligned chords — Proposition 5.1. An attained positive defect gives chords x = ty + pv, z = ty - rv and f(x) = tb + sw, f(z) = tb - uw with s - p = r - u = M and p + r = s + u = ||x - z|| <= 2.
• A secant estimate — Lemma 2.1. For supports phi at v + theta y: eps/p <= phi(v) <= K, with K <= 1 - B if B <= eps and K <= B if B > eps.
• Long half-chords are impossible — Lemma 3.3. In a chord configuration A = p/d and B = u/d are at most eps = 1 - t.
• No chord configuration exists — Proposition 4.1. Convexity of the norm along the two chord directions contradicts the preserved distance between y and z.
• Tingley's problem (OpenAI) — Theorem 5.1. Every surjective isometry between unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry, the radial map.
• Normed spaces — Corollary 5.2. The same holds for nonzero real normed spaces.

Reading order

Read the lessons in order. The first collects supports, radial contact and the Mazur–Ulam theorem and reduces the problem to the defect. The second provides ultrapowers and Darbo's theorem. The third finds aligned chords. The fourth shows that aligned chords cannot exist and proves the theorem.