The second duals of certain spaces of analytic functions

Let ϕ be a continuous, decreasing, real-valued funtion on 0 ≦ r ≦ 1 with ϕ(1) = 0 and ϕ( r ) > 0 for r < 1. Let E 0 be the Banach space of analytic function f on the open unit disc D , such that f ( z )φ(|z|) → 0 as |z| → 1, with norm , where we write ϕ( z ) = ϕ( z ) for z ∈ D . Let E be the Banach space of analytic functions f on D for which f φ is bounded in D , with the same norm as E 0 . It is easy to see that E is complete in this norm, and that E 0 is a closed subspace of E .