PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] absolute value in a vector lattice (Definition)

Let $ V$ be a vector lattice over $ \mathbb{R}$, and $ V^+$ be its positive cone. We define three functions from $ V$ to $ V^+$ as follows. For any $ x\in V$,

  • $ x^+:=x\vee 0$,
  • $ x^-:=(-x)\vee 0$,
  • $ \vert x\vert:=(-x)\vee x$.

It is easy to see that these functions are well-defined. Below are some properties of the three functions:

  1. $ x^+=(-x)^-$ and $ x^-=(-x)^+$.
  2. $ x=x^+-x^-$, since $ x^+-x^-=(x\vee 0)-(-x)\vee 0=(x\vee 0)+(x\wedge 0)=x+0=x$.
  3. $ \vert x\vert=x^++x^-$, since $ x^++x^-=x+2x^-=x+(-2x)\vee 0=(x-2x)\vee (x+0)=\vert x\vert$.
  4. If $ 0\le x$, then $ x^+=x$, $ x^-=0$ and $ \vert x\vert=x$. Also, $ x\le 0$ implies $ x^+=0$, $ x^-=-x$ and $ \vert x\vert=-x$.
  5. $ \vert x\vert=0$ iff $ x=0$. The “only if” part is obvious. For the “if” part, if $ \vert x\vert=0$, then $ (-x)\vee x=0$, so $ x\le 0$ and $ -x\le 0$. But then $ 0\le x$, so $ x=0$.
  6. $ \vert rx\vert=\vert r\vert\vert x\vert$ for any $ r\in \mathbb{R}$. If $ 0\le r$, then $ \vert rx\vert=(-rx)\vee (rx)=r\big((-x)\vee x\big)=r\vert x\vert=\vert r\vert\vert x\vert$. On the other hand, if $ r\le 0$, then $ \vert rx\vert=(-rx)\vee (rx)=(-r)\big(x\vee (-x)\big)=-r\vert x\vert=\vert r\vert\vert x\vert$.
  7. $ \vert x\vert+\vert y\vert=\vert x+y\vert\vee \vert x-y\vert$, since
    $\displaystyle LHS=(-x)\vee x+(-y)\vee y=(-x-y)\vee (-x+y)\vee (x-y)\vee (x+y)=RHS.$
  8. (triangle inequality). $ \vert x+y\vert\le \vert x\vert+\vert y\vert$, since $ \vert x+y\vert\le \vert x+y\vert\vee \vert x-y\vert=\vert x\vert+\vert y\vert$.

Properties 5, 6, and 8 satisfy the axioms of an absolute value, and therefore $ \vert x\vert$ is called the absolute value of $ x$. However, it is not the “norm” of a vector in the traditional sense, since it is not a real-valued function.



"absolute value in a vector lattice" is owned by CWoo.
(view preamble)

View style:

Also defines:  absolute value

This object's parent.

Attachments:
solid set (Definition) by CWoo
Log in to rate this entry.
(view current ratings)

Cross-references: vector, axioms, triangle inequality, iff, implies, properties, well-defined, easy to see, functions, positive cone, vector lattice
There are 26 references to this entry.

This is version 7 of absolute value in a vector lattice, born on 2007-05-07, modified 2008-06-21.
Object id is 9345, canonical name is AbsoluteValueInAVectorLattice.
Accessed 1410 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.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)