normed

Normed describes a mathematical structure in which a norm is defined on a vector space to measure the length of vectors. A normed space consists of a vector space V over the real or complex numbers and a function ||·||: V → [0, ∞) called a norm. The norm must satisfy positivity (||x|| ≥ 0 and ||x|| = 0 only for x = 0), scalability (||αx|| = |α| ||x|| for any scalar α), and the triangle inequality (||x + y|| ≤ ||x|| + ||y|| for all x,y in V). These axioms imply a metric d(x,y) = ||x − y||, which induces notions of convergence and continuity on V.

If every Cauchy sequence in V with respect to this metric converges to a limit in V, the normed space is complete and is called a Banach space. If completeness is not assumed, the structure is simply a normed space.

Examples include the Euclidean space R^n with the Euclidean norm ||x||2 = sqrt(x1^2 + ... + xn^2), as well as the family of L^p spaces with norms ||f||p = (∫ |f|^p)^{1/p} for 1 ≤ p < ∞, and the sup norm ||f||∞ = ess sup |f|. Some norms arise from inner products via ||x|| = sqrt(<x,x>); such norms are compatible with the geometry of the inner product space, though not every norm comes from an inner product.

Norms extend to algebras with submultiplicativity: ||xy|| ≤ ||x|| ||y||, making it possible to study normed algebras and related structures. The operator norm, defined for a linear map T between normed spaces by ||T|| = sup_{||x||=1} ||Tx||, provides a central tool for measuring the size of operators.