vector lattice

An ordered vector space whose underlying poset is a latticeMathworldPlanetmath is called a vector lattice. A vector lattice is also called a Riesz space.

For example, given a topological spaceMathworldPlanetmath X, its ring of continuous functions C(X) is a vector lattice. In particular, any finite dimensional Euclidean space n is a vector lattice.

A vector sublattice is a subspaceMathworldPlanetmath of a vector lattice that is also a sublattice.

Below are some properties of the join () and meet () operationsMathworldPlanetmath on a vector lattice L. Suppose u,v,wL, then

  1. 1.


  2. 2.


  3. 3.

    If λ0, then λuλv=λ(uv)

  4. 4.

    If λ0, then λuλv=λ(uv)

  5. 5.

    If uv, then the converseMathworldPlanetmath holds for 3 and 4

  6. 6.

    If L is an ordered vector space, and if for any u,vL, either uv or uv exists, then L is a vector lattice. This is basically the result of property 2 above.

  7. 7.

    (uv)+w=(u+w)(v+w) (dual of statement 1)

  8. 8.

    uv=-(-u-v) (a direct consequence of statement 4, with λ=-1)

  9. 9.



    (-u)uu and (-u)u-u imply that 2((-u)u)u+(-u)=0, so (-u)u0, which means 0-((-u)u)=u(-u). ∎

  10. 10.

    (ab)+(cd)=(a+c)(a+d)(b+c)(b+d), by repeated application of 1 above.

Remark. The first five properties are also satisfied by an ordered vector space, with the assumptionsPlanetmathPlanetmath that the suprema exist for the appropriate pairs of elements (see the entry on ordered vector space for detail).

Title vector lattice
Canonical name VectorLattice
Date of creation 2013-03-22 17:03:13
Last modified on 2013-03-22 17:03:13
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 06F20
Classification msc 46A40
Synonym Riesz Space
Defines vector sublattice