Norm

Intuition

A norm is an object which can measure the "size" or "length" of a vector.

Formal definition

A norm on a vector space $E$ is a function $N:E\rightarrow \R^{+}$ with the following properties, for all vectors $x,y\in E$ and scalars $a\in \R$:

1. Separation: If $N(x)=0$, then $x=0$.
2. Scaling: $N(ax)=|a|N(x)$.
3. Triangle inequality: $N(x+y)\leq N(x)+N(y)$.

Remarks

• Norms are usually denoted
  $$\|.\|$$For instance, $\|(x,y)\|$ is the norm of vector $(x,y)$ in $\R^{2}$.
• A vector space with a norm is called a normed vector space.