Vector lattice

In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.

Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.

Riesz spaces have wide ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz Spaces. E.g. the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in Mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.

A Riesz space E is defined to be a vector space endowed with a partial order, ≤, that for any x, y, z in E, satisfies:

Every Riesz space is a partially ordered vector space, but not every partially ordered vector space is a Riesz space.

Every element f in a Riesz space, E, has unique positive and negative parts, written f ^± = ±f  ∨ 0. Then it can be shown that, f = f ⁺ − f ⁻ and an absolute value can be defined by | f | = f ⁺ + f ⁻. Every Riesz space is a distributive lattice and has the Riesz decomposition property.

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