Cotype, K-convexity and the approximation property

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).

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. Type, cotype and K-convexity
2. Walsh norms and moment lifts
3. A Fourier coefficient of binary rank two
4. Growth from a slow decrease
5. Uniform decay and the characterization

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
• Tensor products of Banach and Hilbert spaces, Jordan homomorphisms and isometries of \(C^*\)-algebras · Tensor products of operator algebras
• Banach spaces and the Hahn-Banach theorem (course Foundations of von Neumann algebras); injective and projective tensor norms (course Tensor products of operator algebras).