pnormákra

Pnornákra, or p-norms, refer to a family of norms on vector spaces parameterized by p ≥ 1. For a vector x = (x1, ..., xn) in R^n, the p-norm is defined as ||x||_p = (sum_{i=1}^n |x_i|^p)^{1/p}. When p = ∞, ||x||_∞ = max_i |x_i|. In functional analysis this extends to L^p spaces on measure spaces: ||f||_p = (∫ |f|^p)^{1/p} for 1 ≤ p < ∞; ||f||_∞ = ess sup |f|.

These expressions define norms for p ≥ 1; they satisfy positivity, homogeneity, and the triangle inequality (Minkowski). They induce the usual topology on R^n and on L^p spaces. The geometry of unit balls varies with p: p = 1 yields a diamond, p = 2 a circle, p = ∞ a square. In finite dimensions, different p-norms measure distance and magnitudes in distinct ways, influencing optimization and data representations.

Duality and inequalities: For 1 ≤ p ≤ ∞ with 1/p + 1/q = 1, the dual space of ℓ^p is ℓ^q. Hölder's inequality states |⟨x, y⟩| ≤ ||x||_p ||y||_q; Minkowski's inequality gives ||x + y||_p ≤ ||x||_p + ||y||_p. In function spaces, L^p and L^q are dual when 1 ≤ p < ∞.

Variants and applications: For p ∈ (0,1) the formula defines a quasi-norm, not a norm, and it appears in certain nonconvex optimization contexts. In practice, p-norms are used as regularizers in statistics and machine learning: L1 promotes sparsity, L2 promotes small coefficients, and intermediate p values balance these effects. They also arise in signal processing and numerical analysis.