Definition (Norm)

Let $V$ be a $\mathbb{F}$-vector space. A norm on $V$ assigns to each vector $v\in V$ a “magnitude” $\|v\|\in\mathbb{R}^{+}$, and the assignment satisfies the following:

1. Positive Definiteness: $$\|v\|\ge0, \ \forall v\in V$$ or $$\|v\|=0 \iff v=0_{v}$$
2. Homogeneity: $$\|av\|=|a|\cdot\|v\|$$$\forall a\in\mathbb{R}$ and $\forall v\in V$.
3. Triangle Inequality: $$\|v_{1}+v_{2}\|\le\|v_{1}\|+\|v_{2}\|$$$\forall v_{1},v_{2}\in V$.

Definition (Complex norm)

If $z=a+bi$ is a complex number, we define the Norm of z to be the complex number $|z|:=\sqrt{a^2+b^2}$

Definition (1 norm)

The 1-Norm, denoted as $\|\cdot\|$ is defined as follows for any arbitrary $v\in V$ where $v=(v_{1},\ldots,v_{n})$ $$\begin{align*} \mathbb{F}^{n}&: \ \|v\|=|v_{1}|+\cdots+|v_{n}|\\ \mathbb{F}^{\infty}&:\|(v_{i})_{i\in\mathbb{N}}\|=\sum\limits_{i=1}^{\infty}|v_{i}|\\ C^{0}([a,b];\mathbb{F})&:\|f\|=\int_{a}^{b}|f(x)|dx \end{align*}$$

Definition (2 norm)

The 2-Norm, denoted as $\|\cdot\|_{2}$, for any arbitrary $v\in V$ is defined as follows: $$\begin{align*} \mathbb{F}^{n}&: \ \|v\|_{2}=\sqrt{|v_{1}|^{2}+\cdots+|v_{n}|^{2}}\\ \mathbb{F}^{\infty}&:\|(v_{i})_{i\in\mathbb{N}}\|_{2}=\left(\sum\limits_{i=1}^{\infty}|v_{i}|^{2}\right)^{\frac{1}{2}}\\ C^{0}([a,b];\mathbb{F})&:\|f\|_{2}=\left(\int_{a}^{b}|f(x)|^{2}dx\right)^{\frac{1}{2}} \end{align*}$$