ACIT4330-Page/content/Definitions/Norm.md

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