# absolute value in a vector lattice

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 $x\in V$,

• $x^{+}:=x\vee 0$,

• $x^{-}:=(-x)\vee 0$,

• $|x|:=(-x)\vee x$.

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 $x^{+}-x^{-}=(x\vee 0)-(-x)\vee 0=(x\vee 0)+(x\wedge 0)=x+0=x$.

3. 3.

$|x|=x^{+}+x^{-}$, since $x^{+}+x^{-}=x+2x^{-}=x+(-2x)\vee 0=(x-2x)\vee(x+0)=|x|$.

4. 4.

If $0\leq x$, then $x^{+}=x$, $x^{-}=0$ and $|x|=x$. Also, $x\leq 0$ 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 $(-x)\vee x=0$, so $x\leq 0$ and $-x\leq 0$. But then $0\leq x$, so $x=0$.

6. 6.

$|rx|=|r||x|$ for any $r\in\mathbb{R}$. If $0\leq r$, then $|rx|=(-rx)\vee(rx)=r\big{(}(-x)\vee x\big{)}=r|x|=|r||x|$. On the other hand, if $r\leq 0$, then $|rx|=(-rx)\vee(rx)=(-r)\big{(}x\vee(-x)\big{)}=-r|x|=|r||x|$.

7. 7.

$|x|+|y|=|x+y|\vee|x-y|$, since

 $LHS=(-x)\vee x+(-y)\vee y=(-x-y)\vee(-x+y)\vee(x-y)\vee(x+y)=RHS.$
8. 8.

(triangle inequality). $|x+y|\leq|x|+|y|$, since $|x+y|\leq|x+y|\vee|x-y|=|x|+|y|$.

Properties 5, 6, and 8 satisfy the axioms of an absolute value, 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 AbsoluteValueInAVectorLattice 2013-03-22 17:03:16 2013-03-22 17:03:16 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 46A40 msc 06F20 absolute value