Metric for Vector Spaces

Definition

• With $B$ as a Vector Space
• With $d:V\times V\to \mathbb{R}$ as a function
• The function $d$ is a metric if:
  1. $d(v,w)\ge 0,\forall v,w\in V$ (Non-negativity)
  2. $d(v,w)=d(w,v)$ (Commutative/Symmetric)
  3. $d(v,w)=0⟺v=w$ (Zero distance for equality)
  4. $d(u,w)\le d(u,v)+d(v,w)$ (Triangle Inequality)

Implementations

• Euclidean Metric for Reals in R2

Theorems

• 1-Metric