Normm

Normm is a term sometimes used informally to refer to the family of Lp norms on a vector space, where the parameter m labels the order of the norm. In this usage, for a vector x in Rn, the norm is defined as ||x||m = (sum_i |x_i|^m)^(1/m). When m is an element of [1, ∞), this function is a genuine norm, and it is widely known as the Lp norm with p = m. As m increases, the norm becomes more sensitive to the largest coordinate; as m decreases toward 1, it emphasizes a larger number of coordinates.

Special cases are well known: m = 2 yields the Euclidean norm, m = 1 gives the Manhattan (L1) norm, and as m approaches infinity, ||x||∞ equals the maximum absolute value among the components. For 0 < m < 1, the function defined by the same formula is a quasi-norm, not a norm, because it fails the triangle inequality.

Normm/Lp norms have broad applications in mathematics, statistics, and machine learning. They are used to measure vector lengths, to regularize optimization problems, and to promote different kinds of sparsity or smoothness in solutions. In optimization, the choice of m affects differentiability: norms with m > 1 are differentiable away from the origin, while the L1 norm is nondifferentiable at zero.

Although normm is not a formal, universally standardized term, it is commonly understood as shorthand for the Lp family of norms. For precise usage, it is better to refer to the Lp norm with the explicit parameter p. See also Lp norm, quasi-norm, and related inequalities.