Norm

Definition

$\Vert \cdot \Vert :V\to {\mathbb{R}}_0^{+}$ A real-valued function with the following properties

• Positive definiteness: $\forall \mathbf{v}\in V:\Vert \mathbf{v}\Vert =0\iff \mathbf{v}=0$
• Absolute Homogeneity: $\forall \mathbf{v}\in V,k\in K:\Vert k\mathbf{v}\Vert =|k|\Vert \mathbf{v}\Vert$
• Sub additivity or Triange inequality: $\forall \mathbf{v},\mathbf{w}\in V:\Vert \mathbf{v}+\mathbf{w}\Vert \le \Vert \mathbf{v}\Vert +\Vert \mathbf{w}\Vert$ where $K\in \{\mathbb{R},\mathbb{C}\}$ on a vector space $(V,+,\cdot ,K)$

Vector

Real

Let $\mathbf{x}\in {\mathbb{R}}^n$ be an $n$-dimensional vector, then the $L_p$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}{\Vert }_p:={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{1/p}$

Complex

Let $\mathbf{x}\in {\mathbb{C}}^n$ be an $n$-dimensional complex vector, then the $L_1$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}\Vert ={\mathbf{x}}^{†}\mathbf{x}=\sqrt{{\sum }_{i=1}^n{\bar{x}}_i{x}_i}$

Function

$\Vert f\Vert =\sqrt{⟨f,f⟩}=\sqrt{{\int }_a^{b}f(x{)}^{2}dx}$: norm of $f$

Facts

$\Vert \frac{f(x)}{\Vert f\Vert }\Vert =1$

a norm can be induced by a Inner product $d(v,w)=\Vert v-w\Vert$