Results on minimal invariant sets, resolvents, the anchor, the tree and Kirk'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.

• Weak compactness in reflexive spaces — Lemma 1.1. In a reflexive Banach space every closed bounded convex set is weakly compact.
• Reduction to a minimal invariant set — Lemma 2.1. A counterexample yields a separable weakly compact convex K of diameter 1, minimal for a fixed-point-free nonexpansive T, with metrizable weak topology and weakly compact unit ball of the closed span of K - K.
• Diametral approximate fixed points (Goebel, Karlovitz) — Lemma 3.1. Every approximate fixed point sequence in a minimal set K has distance tending to 1 from every point of K.
• Kirk's theorem — Theorem 3.2. In a reflexive space, every nonexpansive self-map of a closed bounded convex set with normal structure has a fixed point.
• Compact detector — Lemma 3.3. For compact E in K and eps > 0, every sufficiently good approximate fixed point y has one functional of norm at most 1 with f(y - z) >= 1 - eps for all z in E.
• Resolvents — Lemma 1.1. R_n(a) = a/n + (1 - 1/n) T R_n(a) defines a nonexpansive R_n with ||T R_n a - R_n a|| <= 1/n and ||R_n a - a|| <= (n - 1)||Ta - a||.
• Idempotents in compact semigroups (Ellis) — Lemma 4.1. A nonempty compact Hausdorff semigroup with continuous right translations contains an idempotent.
• The anchor point — Proposition 5.1. The intersection of the pointwise weak closures of the convex hulls of the tail families is a compact semigroup with an idempotent P; a fixed point x of P lies in the closed convex hull of the weak cluster set of its tail images.
• Adaptive selection — Lemma 6.1. Maps G_i chosen one at a time from tail families with adaptively prescribed precision admit, after finitely many steps, a convex combination of the G_i(x) within any given tau of x.
• Evaluation on an ordered tree — Lemma 1.1. Path weights sum to 1 on each level and to W_v on each block; the root output is W-weighted Lipschitz in the outputs prescribed on a level.
• Compact families of predictions — Lemma 2.2. The weak ultrafilter limits Z_v(theta) of the predicted root outputs are norm Lipschitz in theta, so E_v is norm compact and known before v is chosen.
• The tree — Proposition 3.1. An infinite ordered tree with summable budgets e_v, child averages within tau_k of x, ancestor displacements at most e_v, and functionals with f_v(y_v - z) >= 1 - e_v on E_v.
• Ideal cutoffs — Lemma 4.1. The root output of the ideal cutoff state below v equals the prediction Q_v(b; theta^v).
• Detecting a block — Lemma 2.1. When the leaves below v switch to a common point b, the root output moves by U_v - V_v with f_v(U_v - V_v) >= W_v - 5 e_v.
• A finite witness — Proposition 3.1. At every depth some leaf s and some d in the unit ball of Y have f_v(d) >= 1/2 for all nonroot v on the path to s.
• Kirk's problem (OpenAI) — Theorem 4.1. Every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point.
• The fixed point property of reflexive spaces — Corollary 4.2. Every reflexive Banach space has the fixed point property for nonexpansive maps; equivalently for weakly compact convex domains.

Reading order

Read the lessons in order. The first reduces the problem to a minimal invariant set and proves the diametral lemma. The second constructs the anchor point and the adaptive selection. The third builds the tree. The fourth runs the switching experiment and proves the theorem.