pNormen

p-normen, in English p-norms, refer to a family of norms on vector spaces defined by a parameter p ≥ 1. In R^n, the p-norm of a vector x = (x1, ..., xn) is ||x||_p = (∑i |xi|^p)^{1/p}. The concept extends to function spaces such as L^p spaces on measure spaces, where ||f||_p = (∫ |f|^p dμ)^{1/p}.

Special cases include p = 2, the Euclidean norm; p = 1, the L1 or Manhattan norm; and p = ∞, the maximum norm, defined as ||x||_∞ = max_i |xi| (or ess sup |f| in function spaces). The limit as p → ∞ yields the max norm.

Properties: For p ≥ 1, ||·||_p satisfies positivity, homogeneity, and the triangle inequality (Minkowski). In finite-dimensional spaces, all p-norms are equivalent: for 1 ≤ p ≤ q ≤ ∞ and x ∈ R^n, ||x||_q ≤ ||x||_p ≤ n^{(1/p - 1/q)} ||x||_q.

Infinite-dimensional spaces: sequences in l^p form a normed space with ||x||_p = (∑ |x_i|^p)^{1/p}. Duality: for 1 ≤ p < ∞, the continuous dual of l^p is l^q, where 1/p + 1/q = 1. The case p = ∞ has a larger dual than l^1.

Applications: p-norms are fundamental in analysis, optimization, statistics, and machine learning. L1 regularization promotes sparsity, L2 regularization emphasizes smoothness, and different p-norms serve as loss functions, constraints, or measures of deviation in various algorithms.

Variants: for 0 < p < 1, ∥·∥_p is not a norm but a quasi-norm, used in sparse representations and certain signal-processing contexts.