Norms

Definition: Norm

A norm on a complex or a real vector space $(V,F,+,\cdot )$, where $F$ is $\mathbb{C}$ or $\mathbb{R}$, is any function $N:V\to \mathbb{R}$ with the following properties:

• $N(\mathbf{v})\ge 0$ and $N(\mathbf{v})=0⟺\mathbf{v}=0$ for all $\mathbf{v}\in V$
• $N(\lambda \mathbf{v})=|\lambda |\cdot N(\mathbf{v})$ for all $\lambda \in F,\mathbf{v}\in V$
• $N(\mathbf{u}+\mathbf{v})\le N(\mathbf{u})+N(\mathbf{v})$ for all $\mathbf{u},\mathbf{v}\in V$ (triangle inequality)

Definition: Normed Vector Space

A normed vector space is a vector space equipped with a norm.

Definition: Unit Vector

A unit vector in a normed vector space is any vector whose norm is equal to $1$.

Notation

Unit vectors are usually denoted with hats: $\hat{\mathbf{u}},\hat{\mathbf{v}}$, etc.