{
  "id": "lipschitz-free-spaces-and-bounded-approximation",
  "title": "Lipschitz-free spaces and bounded approximation",
  "description": "The Lipschitz-free space of a metric space, and the approximation property and the bounded approximation property of Banach spaces. The theorem: there is a countable metric space with all distances at least 1 whose free space has the approximation property and not the bounded one; and the space ℓ₁ has equivalent norms without the metric approximation property. The proof: bounded blocks glued at a point; graphs of large degree whose small balls look like a tree with independent colors; signed measures of an operator; walks that can be followed by finitely many points; test functions on the edges and an average. The main theorems need no hypothesis; one consequence rests on a named theorem of Kalton. The course is an exposition of OpenAI's construction (2026).",
  "units": [
    {
      "id": "lipschitz-free-spaces-and-approximation-properties",
      "title": "Lipschitz-free spaces and approximation properties",
      "reader": "lipschitz-free-spaces-and-approximation-properties.html",
      "source": "src/lipschitz-free-spaces-and-approximation-properties.md",
      "source_sha256": "1741a3307bb8a39b2a74936540a3e43f32a0bfc6e7e1cccd402cd4b59b8e6b51",
      "status": "author_self_checked"
    },
    {
      "id": "a-wedge-of-bounded-blocks",
      "title": "A wedge of bounded blocks",
      "reader": "a-wedge-of-bounded-blocks.html",
      "source": "src/a-wedge-of-bounded-blocks.md",
      "source_sha256": "0f8328bb64d7ee5e99656778613113e4863b5f1e1145d22ad6c38551294b9a54",
      "status": "author_self_checked"
    },
    {
      "id": "colored-graphs-with-a-tree-limit",
      "title": "Colored graphs with a tree limit",
      "reader": "colored-graphs-with-a-tree-limit.html",
      "source": "src/colored-graphs-with-a-tree-limit.md",
      "source_sha256": "c08528b31f4ae7fc4424ca784667ca5cb4252db675e2a59f97ed5d15cd549dff",
      "status": "author_self_checked"
    },
    {
      "id": "signed-measures-and-matched-walks",
      "title": "Signed measures and matched walks",
      "reader": "signed-measures-and-matched-walks.html",
      "source": "src/signed-measures-and-matched-walks.md",
      "source_sha256": "b430fba623efbce233d3c0fda6588915ffdc638317a4b3fbf56eb01d47766bff",
      "status": "author_self_checked"
    },
    {
      "id": "scalar-tests-and-the-anchor-obstruction",
      "title": "Scalar tests and the anchor obstruction",
      "reader": "scalar-tests-and-the-anchor-obstruction.html",
      "source": "src/scalar-tests-and-the-anchor-obstruction.md",
      "source_sha256": "604e8f0bd657a7fca1b13f785f2af6eeb02af2cc8b7b6b1099520002ff17981b",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Functional Analysis (D20): normed spaces, the dual of a normed space and its completeness, bounded operators, compact sets; the Hahn-Banach theorem (second lesson, Section 6 only)",
    "finite probability: uniform random permutations, independence, Chebyshev's inequality (third lesson)",
    "finite graphs and metric spaces",
    "the lessons prove every other fact they use, except the named hypothesis (K) of the second lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}