# Notations and Conventions

## Abstract

By ℕ, ℤ, ℝ and ℂ we denote the natural, integer, real and complex numbers, respectively. We let ℕ_{o} ≔ 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), B_{X} = {x ∈ X | ‖x‖_{X} ≤ 1} and write Id = Id_{X} 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 <x*,x> or x*(x) where x ∈ X, x* ∈ X*. The dual of an operator T is denoted by T*.