# Definition
A **norm** on $V$ is a map $\|\cdot\| : V \to [ \, 0, \infty \rangle$ such that
1. $\| u + v \| \leq \| u \| + \| v \|, \; \forall u,v \in V$
2. $\| c \times v \| = | c | \times \| v \|, \; \forall v \in V, \, \forall c \in \mathbb{C}$
3. $\| v \| = 0 \implies v = 0$

Think of $\| v \|$ as the length of $v$.
## Norm of 0
$$\| 0 \| = \| 0 \times u \| = \| c \times u \| = \| 0 \| \times \|u \| = 0 \times \| u \| = 0$$