PlanetMath: vector lattice

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

vector lattice

(Definition)

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

For example, given a topological space , its ring of continuous functions is a vector lattice. In particular, any finite dimensional Euclidean space $\mathbb{R}^n$ is a vector lattice.

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

Below are some properties of the join ( $\vee$ ) and meet ( $\wedge$ ) operations on a vector lattice . Suppose $u,v,w\in L$ , then

$(u+w) \vee (v+w)=(u\vee v)+w$
$u\wedge v=(u+v)-(u\vee v)$
If $\lambda\ge 0$ , then $\lambda u\vee \lambda v=\lambda(u\vee v)$
If $\lambda\le 0$ , then $\lambda u\vee \lambda v=\lambda(u\wedge v)$
If $u\ne v$ , then the converse holds for 3 and 4
If is an ordered vector space, and if for any $u,v\in L$ , either $u\vee v$ or $u\wedge v$ exists, then is a vector lattice. This is basically the result of property 2 above.
$(u\wedge v)+w=(u+w)\wedge (v+w)$ (dual of statement 1)
$u\wedge v=-(-u\vee -v)$ (a direct consequence of statement 4, with $\lambda=-1$ )
$(-u)\wedge u\le 0\le (-u)\vee u$
Proof. $(-u)\wedge u\le u$ and $(-u)\wedge u\le -u$ imply that $2((-u)\wedge u)\le u+(-u)=0$ , so $(-u)\wedge u\le 0$ , which means $0\le -((-u)\wedge u)=u\vee (-u)$ . $\qedsymbol$
$(a\vee b)+(c\vee d)=(a+c)\vee (a+d)\vee (b+c)\vee (b+d)$ , by repeated application of 1 above.

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

"vector lattice" is owned by CWoo.

(view preamble | get metadata)

Other names:

Riesz Space

Also defines:

vector sublattice

This object's parent.

Attachments:: absolute value in a vector lattice (Definition) by CWoo
topological vector lattice (Definition) by CWoo

Log in to rate this entry.
(view current ratings)

Cross-references: application, imply, consequence, converse, operations, meet, join, properties, sublattice, subspace, Euclidean space, finite dimensional, ring of continuous functions, topological space, lattice, poset, ordered vector space
There are 4 references to this entry.

This is version 4 of vector lattice, born on 2007-05-06, modified 2007-05-08.
Object id is 9344, canonical name is VectorLattice.
Accessed 2301 times total.

Classification:

AMS MSC:	06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
	46A40 (Functional analysis :: Topological linear spaces and related structures :: Ordered topological linear spaces, vector lattices)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact