Normed Riesz Spaces and Banach Lattices

Abstract

Let V be a real or complex vector space and assume that to each element f ∈ V there is assigned a real number ‖ f ‖ such that

• (i) ‖ f ‖ ≥ 0 for every f ∈ V and ‖ f ‖ = 0 if and only if f = 0,

• (ii) ‖ λf ‖=∣λ ∣ ∙ ‖ f ‖ for every f ∈ V and every (real or complex) number λ,

• (iii) ‖ f + g ‖ ≤ ‖ f ‖ + ‖ g‖ for all f and g in V (triangle inequality).