The super order dual of an ordered vector space and the Riesz–Kantorovich formula

Abstract

A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L L is a vector lattice and f f and g g are order bounded linear functionals on L L , then their supremum (least upper bound) f ∨ g f\lor g exists in L ∼ L^\sim and for each x ∈ L + x\in L_+ it satisfies the so-called Riesz–Kantorovich formula: \[ [ f ∨ g ] ( x ) = sup { f ( y ) + g ( z ) : y , z ∈ L + a n d y + z = x } . \bigl [f\lor g\bigr ](x)=\sup \bigl \{f(y)+g(z)\colon \ y,z\in L_+\ \,\mathrm {and}\ \, y+z=x\bigr \}\,. \] Related to the Riesz–Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f f and g g on an ordered vector space exists, does it then satisfy the Riesz–Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz–Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz–Kantorovich formula.