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 \(\A, \B \in \real^{n \times n}\)