OAI.Cotype.mainTarget_proved

The theorem states that, for every real Banach space X (a complete real normed space) that is nontrivial, the defined proposition MainTarget(X) holds. MainTarget(X) says: if X has the approximation property, then X is K-convex if and only if both X and its dual space of continuous linear functionals X →L[ℝ] ℝ have finite cotype. Here the approximation property means that for every compact set M ⊆ X and every δ > 0 there is a continuous linear operator S : X → X with finite-dimensional range such that ‖Sx − x‖ < δ for all x in M. On the discrete cube {±1}ⁿ (with sign false = +1 and true = −1), the L² norm of f : cube → X is the square root of the average of ‖f(ε)‖² over all ε. The moment of f at coordinate i is the average of ε_i f(ε), and the Rademacher projection of f is the function ε ↦ Σᵢ ε_i times the moment at i. X is K-convex if there is a constant K ≥ 0 such that, for every n and every f on the n-cube, the L² norm of the Rademacher projection of f is at most K times the L² norm of f. X has cotype q, for real q ≥ 2, if there is C ≥ 0 such that, for every n and all vectors x₁, …, xₙ in X, (Σ ‖xᵢ‖^q)^(1/q) ≤ C times the L² norm of ε ↦ Σ ε_i xᵢ. Finite cotype means cotype q for some such q.