Definition

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 $|z|:=\sqrt{a^2+b^2}$