READING MATHEMATICS
Expository notes for following an argument in detail.
These notes explain the ideas, background and logical dependencies of selected proofs. Each note identifies its original source and offers a route through the details.
GEOMETRY OF NORMED SPACES
A reader’s reconstruction of OpenAI’s argument, with the convex invariant-set construction and the final chord-coordinate comparison developed step by step.
Introduction, reading route and PDFFIXED POINTS IN BANACH SPACES
A detailed reconstruction through ordered trees, root responses and finite switching, with an RNP extension and no priority claim.
Introduction, reading route and PDFBROUWER–SCHAUDER FIXED POINTS
An operator-and-polynomial proof of Brouwer's fixed-point theorem, followed by a direct passage from the cube theorem to Schauder's compact-image theorem.
Introduction, reading route and PDFLIPSCHITZ-FREE SPACES
A detailed introduction to Lipschitz-free spaces, from linearization and compact reduction to the absence of c0 in the free space over a real Hilbert space.
Introduction, reading route and PDFLIPSCHITZ HOMEOMORPHISMS
A Lipschitz homeomorphism preserves distances up to multiplicative constants, but it need not preserve the linear structure of a Banach space. This Expository Note reconstructs OpenAI's counterexample for separable real Banach spaces and explains how the construction fits together, from Lipschitz-free-space inputs to weighted graph spaces and an explicit nonlinear change of variables.
Introduction, reading route and PDF