absolute value in a vector lattice (original) (raw)

Let V be a vector lattice over ℝ, and V+ be its positive cone. We define three functions from V to V+ as follows. For any x∈V,

• •
  x+:=x∨0,
• •
  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. 1. x=x+-x-, since x+-x-=(x∨0)-(-x)∨0=(x∨0)+(x∧0)=x+0=x.
3. 1. |x|=x++x-, since x++x-=x+2⁢x-=x+(-2⁢x)∨0=(x-2⁢x)∨(x+0)=|x|.
4. 1. If 0≤x, then x+=x, x-=0 and |x|=x. Also, x≤0 implies x+=0, x-=-x and |x|=-x.
5. 1. |x|=0 iff x=0. The “only if” part is obvious. For the “if” part, if |x|=0, then (-x)∨x=0, so x≤0 and -x≤0. But then 0≤x, so x=0.
6. 1. |r⁢x|=|r|⁢|x| for any r∈ℝ. If 0≤r, then |r⁢x|=(-r⁢x)∨(r⁢x)=r⁢((-x)∨x)=r⁢|x|=|r|⁢|x|. On the other hand, if r≤0, then |r⁢x|=(-r⁢x)∨(r⁢x)=(-r)⁢(x∨(-x))=-r⁢|x|=|r|⁢|x|.
7. 1. |x|+|y|=|x+y|∨|x-y|, since

      L⁢H⁢S=(-x)∨x+(-y)∨y=(-x-y)∨(-x+y)∨(x-y)∨(x+y)=R⁢H⁢S.