{
  "id": "cotype-k-convexity-and-the-approximation-property",
  "title": "Cotype, K-convexity and the approximation property",
  "description": "A nonzero real Banach space with the approximation property is K-convex if and only if both the space and its dual have finite Rademacher cotype; without the approximation property this fails. The proof: Walsh norms of operators, Kahane's inequality and moment lifts; a Fourier coefficient of binary rank two for functions of sign matrices; growth of Walsh norms from a slow decrease, through frames over F_2; uniform decay for finite-rank operators; a compact domain that carries the decay to the identity through the approximation property. The decay theorem is proved in full; two classical theorems of Pisier are named hypotheses. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "type-cotype-and-k-convexity",
      "title": "Type, cotype and K-convexity",
      "reader": "type-cotype-and-k-convexity.html",
      "source": "src/type-cotype-and-k-convexity.md",
      "source_sha256": "d08f9f6d118680deafb30df884c459e653d2cc7bf1c677ce0f0d01b8d2f19ca0",
      "status": "author_self_checked"
    },
    {
      "id": "walsh-norms-and-moment-lifts",
      "title": "Walsh norms and moment lifts",
      "reader": "walsh-norms-and-moment-lifts.html",
      "source": "src/walsh-norms-and-moment-lifts.md",
      "source_sha256": "61b78c179420855f4b4308fb442930038f91af5e330819eac178546666e5a3db",
      "status": "author_self_checked"
    },
    {
      "id": "a-fourier-coefficient-of-binary-rank-two",
      "title": "A Fourier coefficient of binary rank two",
      "reader": "a-fourier-coefficient-of-binary-rank-two.html",
      "source": "src/a-fourier-coefficient-of-binary-rank-two.md",
      "source_sha256": "59f699d9b3e8b812741b8e80d9e99b0fc3e97052572cab98edfd95b027d22261",
      "status": "author_self_checked"
    },
    {
      "id": "growth-from-a-slow-decrease",
      "title": "Growth from a slow decrease",
      "reader": "growth-from-a-slow-decrease.html",
      "source": "src/growth-from-a-slow-decrease.md",
      "source_sha256": "4c7b6abd5a4caed679622b17e71e7f24025a8ef9331ae2f3f6371b341475e887",
      "status": "author_self_checked"
    },
    {
      "id": "uniform-decay-and-the-characterization",
      "title": "Uniform decay and the characterization",
      "reader": "uniform-decay-and-the-characterization.html",
      "source": "src/uniform-decay-and-the-characterization.md",
      "source_sha256": "c543388d1b589b05fbfd12f2cb638bf7009e71fda65ca9dca8a31f3b876900bd",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Foundations of von Neumann algebras, lesson Hahn-Banach, Baire and the basic theorems on Banach spaces: the Hahn-Banach theorem for normed spaces (Corollary 2.3)",
    "course Tensor products of operator algebras, lesson Tensor products of Banach and Hilbert spaces, Jordan homomorphisms and isometries of C*-algebras: the injective and projective norms and their metric mapping property (Proposition 2.4)",
    "finite-dimensional linear algebra over the reals and over F_2; Fourier analysis on finite sign cubes (proved in the third lesson); compactness in finite dimensions",
    "the lessons prove every other fact they use, except the named hypotheses (P) and (MP) of the first lesson and (GMR) and (FTJ) of the last section"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}