Norm

This is a Function that takes a element within a Integral Domain to convert it into its length/size representation within a Field.

Definition

• $f:D\to \mathbb{R}$ s.t
• $f(x)>0$
• if $f(x)\ne 0\wedge f(y)\ne 0⟹f(x)\le f(x)f(y)$

Properties

With $\mathcal{D}$ as an Integral Domain or Field

1. $|x|\ge 0,\forall x\in \mathcal{D}$ (Non-negativity)
2. $|x|=0⟺x=0$
3. $|xy|=|x||y|,\forall x,y\in \mathcal{D}$ (Multiplication)
4. $|x+y|\le |x|+|y|,\forall x,y\in \mathcal{D}$ (Triangle Inequality)

Proofs

• Proving Normal Operator Property 4

Implementations

• Absolute Function for Real Numbers
• Absolute Function for Complex Numbers
• Absolute Value for Finite Field
• Absolute Function for Polynomials

Theorems

• Absolute Value of Multiplicative Identity is 1
• All Vector Norms Induce a Metric Theorem
• Nice Metrics Induce a Norm Theorem