Order Duals and Adjoint Operators

Abstract

According to Corollary 20.3 the linear functional φ on the Riesz space E is order bounded if and only if φ is regular, i.e., if and only if φ = φ_l - φ₂ with φ₁ and φ₂ positive. The vector space E~ of all order bounded linear functionals on E is a Dedekind complete Riesz space with respect to the ordering defined by saying that φ_l ≤ φ₂ holds whenever φ₂ - φ₁ is positive. It follows that for any φ ∈ E~ the functionals φ⁺ = φ ∨ 0 and φ^- = (-φ) ∨ 0 exist in E_~ and φ = φ⁺ - φ _-. Also, | φ |= φ ∨ (-φ) exists in E~ and | φ |= φ⁺ + φ^-. The space E_~ is called the order dual of E (see Corollary 20.3).