An Approximation Property and the Sphere of L1

A normed linear space, E, has property P if for every ε>0 there is a δ>0 such that for all x∈E there is a z, with |z|⩽ε, such that B[0, 1+δ]∩B[x, 1]⊆B[z, 1]. This paper contains a simple proof that an infinite dimensional L1(X, Σ, μ) space does not satisfy property P.