Norm

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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 \}


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