Definition: A norm is a function $\norm{\cdot}: \real^d \to \real$ such that for all $\x,\y \in \real^d$,

• Positivity: $\norm{\x} \geq 0$, with $\norm{\x} = 0$ iff $\x=\zero$
• Homogeneity: $\norm{a\x} = \abs{a}\norm{\x}$ for any $a \in \real$
• Triangle inequality: $\norm{\x+\y} \leq \norm{\x} + \norm{\y}$

Definition: A norm is a function $\norm{\cdot}: \real^{m \times n} \to \real$ such that for all $\A, \B \in \real^{m \times n}$,

• Positivity: $\norm{\A} \geq 0$, with $\norm{\A} = 0$ iff $\A=\zero$
• Homogeneity: $\norm{c\A} = \abs{c}\norm{\A}$ for any $c \in \real$
• Triangle inequality: $\norm{\A+\B} \leq \norm{\A} + \norm{\B}$
• Submultiplicativity: $\norm{\A\B} \leq \norm{\A} \norm{\B}$ for all $\A, \B \in \real^{n \times n}$