A.3.2 Special Norms

The EUCLIDean^A.1 norm for a vector ${\mathbf{{x}}}\in \mathbb{R}^n$ is defined as

$$\Vert{\mathbf{{x}}}\Vert _2 {=}\sqrt{ \sum_{i=1}^n \vert x_i\vert^2}.$$ (A.8)

The square of the EUCLIDean norm of a vector ${\mathbf{{x}}}\in \mathbb{R}$ can be written as

$$\Vert{\mathbf{{x}}}\Vert _2^2 {=}\sum_{i=1}^n \vert x_i\vert^2 {=}{\mathbf{{x}}}^{\mathcal{T}}{\mathbf{{x}}}.$$ (A.9)

More generally, the $p$ -norm of the same vector ${{\mathbf{{x}}}}$ is defined as

$$\Vert{\mathbf{{x}}}\Vert _p {=}\sqrt[p]{ \sum_{i=1}^n \vert x_i\vert^p}.$$ (A.10)

The maximum norm of a vector is defined as

$$\lim_{p\rightarrow\infty} \Vert{\mathbf{{x}}}\Vert _p {=}\Vert{\mathbf{{x}}}\Vert _{\infty} {=}\max\{\vert x_i\vert\}.$$ (A.11)