Norm

linear-algebra

Definition

Norm

A norm is a function that assigns a strictly positive length or magnitude to each vector in a vector space - except the zero vector, which has a length of 0.

Properties

Non-Negativity

$$\begin{array}{rl}||\mathbf{v}||& \ge 0\forall \mathbf{v}\in V\setminus \{0\}\\ ||\mathbf{v}||& =0⟺\mathbf{v}=0\end{array}$$

where $0$ is the zero vector.

Absolute Homogeneity

$$||\lambda \cdot \mathbf{v}||=|\lambda |\cdot ||\mathbf{v}||\forall \mathbf{v}\in V,\lambda \in \mathbb{R}$$

Triangle Inequality

$$||\mathbf{u}+\mathbf{v}||\le ||\mathbf{u}||+||\mathbf{v}||\forall \mathbf{u},\mathbf{v}\in V$$