Vector Space Norm

A norm on a vector space $V$ over $\mathbb{F}$, where $\mathbb{F}$ is the real or complex numbers, is a map $\parallel \cdot \parallel :V\to \mathbb{R}$ such that for any $\mathit{u},\mathit{v}$ in $V$ and $\lambda$ in $\mathbb{F}$ the following properties hold:

1. Positive Definite: $||\mathit{v}||\ge 0$ with $||\mathit{v}||=0$ if and only if $\mathit{v}=\mathbf{0}$.
2. Preserve Scalar Multiplication: $||\lambda \mathit{v}||=|\lambda |||\mathit{v}||$
3. Triangle Inequality: $||\mathit{v}+\mathit{u}||\le ||\mathit{v}||+||\mathit{u}||$.