Normerade

Normerade is the Swedish term used in mathematics to refer to normed spaces. A normed space is a vector space V over the real or complex numbers equipped with a norm ||·||: V → [0, ∞) that satisfies positivity, definiteness (||x|| = 0 iff x = 0), homogeneity (||αx|| = |α|·||x|| for all scalars α), and the triangle inequality (||x + y|| ≤ ||x|| + ||y||). The norm induces a metric d(x, y) = ||x − y||, which defines notions of distance, convergence, and continuity on V.

Examples include R^n with the Euclidean norm ||x||2, and function spaces such as L^p spaces on a measure space with ||f||p = (∫|f|^p)1/p for p ≥ 1, or the space of bounded functions with the sup norm ||f||∞. In finite-dimensional spaces, all norms are equivalent in the sense that they induce the same notions of convergence and continuity, and such spaces are necessarily complete.

A normed space that is complete with respect to its norm-derived metric is called a Banach space. Normed spaces provide the framework for much of functional analysis, including the study of linear operators, duality, and various convergence concepts. They also relate to broader categories such as inner product spaces and Hilbert spaces, where the norm arises from an inner product.

History and terminology: the concept of a norm evolved in the early 20th century, with Banach spaces playing a central role in the development of functional analysis. The term normed space reflects the central role of the norm in defining geometric and topological structure on a vector space.

See also: norm, Banach space, Lp spaces, Hilbert space, functional analysis.