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 \(N(\mathbf{v})=0\iff \mathbf{v}=\mathbf{0}\) 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 \(1\).