# Norm

This is a function of a vector, say $LaTeX: ||x||,$ that satisfies three properties:

1. Homogeneous: $LaTeX: \textstyle ||tx|| = |t| ||x|| \mbox{ for all (scalars)}, t.$
2. Positive: $LaTeX: \textstyle ||x|| > 0 \mbox{ for } x \ne 0.$ (Note: $LaTeX: ||0|| = 0$ by homogeneity, so $LaTeX: 0$ is the unique vector with zero norm.)
3. Subadditive: $LaTeX: \textstyle ||x + y|| \le ||x|| + ||y||.$

Norms that arise frequently in mathematical programming are:

Euclidean norm (on $LaTeX: \mathbb{R}^n$): $LaTeX: \textstyle ||x|| = \sqrt{\sum_j x_j^2}$

L_inf (on $LaTeX: \mathbb{R}^n$): $LaTeX: \textstyle ||x|| = \max_j\{|x_j|\} (= \lim L_p \mbox{ as } p \to \infty)$

L_p $LaTeX: \textstyle (\mbox{on } \mathbb{R}^n, \mbox{ for } p \ge 1): ||x|| = [\sum_j |x_j|^p]^{1/p}$

Matrix norm (induced by vector norm): $LaTeX: \textstyle ||A|| = \max \left \{||Ax||: ||x||=1 \right \}$

Supremum norm (on function space): $LaTeX: \textstyle ||F|| = \sup \left \{|F(x)|: x \in X \right \}$