Lipschitz-free spaces and bounded approximation

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).

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Lipschitz-free spaces and approximation properties
  2. A wedge of bounded blocks
  3. Colored graphs with a tree limit
  4. Signed measures and matched walks
  5. Scalar tests and the anchor obstruction

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites