Definition 10.1.1 (Norm). Let $E$ be a vector space over $K \in \RC$ and $\norm{\cdot}_{E}: E \to [0, \infty)$, then $\norm{\cdot}_{E}$ is a norm if:

1. For any $x \in E$, $\norm{x}_{E} = 0$ if and only if $x = 0$.

2. For any $x \in E$ and $\lambda \in K$, $\norm{\lambda x}_{E} = \abs{\lambda}\norm{x}_{E}$.

3. For any $x, y \in E$, $\norm{x + y}_{E} \le \norm{x}_{E} + \norm{y}_{E}$.