OAI.DiscreteFree.real_l1_quantitative_renorming

The theorem states that, for every natural number p ≥ 1, there is an additive group norm N on the real ℓ¹ sequence space (the space of real sequences indexed by ℕ with finite sum of absolute values, with its usual ℓ¹ norm ‖·‖) such that N is absolutely homogeneous, N(r·c) = |r|·N(c) for all real r and all c, and N is equivalent to the ℓ¹ norm with explicit constants: (1/2)‖c‖ ≤ N(c) ≤ B(p)‖c‖ for all c, where B(p) = 300p² + 150p + 5. Moreover, when the space is given the norm N while keeping its original real vector-space operations, N makes it a normed real vector space (with the same module structure), it is complete, and it has the bounded approximation property with constant 2B(p) but not with constant p; in particular, when p = 1 it fails the bounded approximation property with constant 1. Here the bounded approximation property with constant Λ means that for every compact set C and every ε > 0 there is a continuous linear operator T on the space with finite-dimensional range and operator norm at most Λ, and a number η < ε, such that ‖Tu − u‖ ≤ η for all u in C, with norms taken in the renormed space.