l2norms

The L2 norm, also known as the Euclidean norm, assigns a length to a vector. For x in R^n with components x1, ..., xn, it is defined as ||x||2 = sqrt(sum_{i=1}^n x_i^2). More generally, in any inner product space, ||x||2 = sqrt(<x, x>).

For functions, the L2 norm generalizes to square-integrable functions. If f is measurable on a measure space (Ω, Σ, μ), the L2 norm is ||f||2 = (∫_Ω |f|^2 dμ)^{1/2}. The space L2(Ω) consists of all such functions and forms a Hilbert space with inner product ⟨f, g⟩ = ∫_Ω f ḡ dμ; ||f||2 = sqrt(⟨f, f⟩). In finite dimensions, these definitions agree with the Euclidean norm.

Key properties include non-negativity, positive definiteness (||x||2 = 0 if and only if x = 0), homogeneity (||αx||2 = |α| ||x||2), and the triangle inequality (||x + y||2 ≤ ||x||2 + ||y||2). The norm is induced by the inner product, since ||x||2^2 = ⟨x, x⟩. It is differentiable for x ≠ 0 and generates the standard Euclidean geometry.

Distance and geometry: the L2 distance between x and y is ||x − y||2. If x and y are orthogonal, ||x + y||2^2 = ||x||2^2 + ||y||2^2 (the Pythagorean theorem).

Applications: L2 norms are widely used in optimization and statistics, including ridge (L2) regularization, machine learning, and signal processing. They are smooth and strictly convex in the variable, supporting efficient gradient-based methods, and they underpin Fourier and functional-analytic frameworks as the defining norm of L2 spaces.