Notations and Conventions

Abstract

By ℕ, ℤ, ℝ and ℂ we denote the natural, integer, real and complex numbers, respectively. We let ℕo ≔ ℕ ∪ {0}. Banach spaces over the field K ∈ {ℝ,ℂ} of real or complex numbers will be abbreviated by the letters X,Y,Z, operators between such spaces by R,S,T. By operators we always mean continuous linear operators between Banach spaces. The (Banach) space of continuous linear operators T from X to Y under the operator norm ‖T‖:= sup {‖Tx‖Y | ‖x‖X = 1} is denoted by L(X,Y). We let L(X) = L(X,X), BX = {x ∈ X | ‖x‖X ≤ 1} and write Id = IdX for the identity map on X. If A is a subset of X, [A] or span A denotes the closed linear hull of A in X. The topological dual of X is denoted X*(= L(X, K)) and the duality pairing written or x*(x) where x ∈ X, x* ∈ X*. The dual of an operator T is denoted by T*.