absolute value in a vector lattice

Let V be a vector lattice over , and V+ be its positive conePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. We define three functions from V to V+ as follows. For any xV,

  • x+:=x0,

  • x-:=(-x)0,

  • |x|:=(-x)x.

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

  1. 1.

    x+=(-x)- and x-=(-x)+.

  2. 2.

    x=x+-x-, since x+-x-=(x0)-(-x)0=(x0)+(x0)=x+0=x.

  3. 3.

    |x|=x++x-, since x++x-=x+2x-=x+(-2x)0=(x-2x)(x+0)=|x|.

  4. 4.

    If 0x, then x+=x, x-=0 and |x|=x. Also, x0 implies x+=0, x-=-x and |x|=-x.

  5. 5.

    |x|=0 iff x=0. The “only if” part is obvious. For the “if” part, if |x|=0, then (-x)x=0, so x0 and -x0. But then 0x, so x=0.

  6. 6.

    |rx|=|r||x| for any r. If 0r, then |rx|=(-rx)(rx)=r((-x)x)=r|x|=|r||x|. On the other hand, if r0, then |rx|=(-rx)(rx)=(-r)(x(-x))=-r|x|=|r||x|.

  7. 7.

    |x|+|y|=|x+y||x-y|, since

  8. 8.

    (triangle inequality). |x+y||x|+|y|, since |x+y||x+y||x-y|=|x|+|y|.

Properties 5, 6, and 8 satisfy the axioms of an absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath, and therefore |x| 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.

Title absolute value in a vector lattice
Canonical name AbsoluteValueInAVectorLattice
Date of creation 2013-03-22 17:03:16
Last modified on 2013-03-22 17:03:16
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 46A40
Classification msc 06F20
Defines absolute value