normruimten

Normruimten, ofte kalt normed spaces, are mathematical structures consisting of a vector space V over the real or complex numbers together with a norm ||·||: V → [0, ∞). The norm satisfies positivity (||x|| ≥ 0, with equality only for x = 0), homogeneity (||αx|| = |α| ||x||), and the triangle inequality (||x + y|| ≤ ||x|| + ||y||). The norm induces a metric d(x, y) = ||x − y||, turning V into a metric space; if every Cauchy sequence converges in V, the normruimten is complete and is called a Banach space.

Typical examples include: the Euclidean space R^n with the Euclidean norm ||x||2 = sqrt(x1^2 + … + xn^2); more generally the p-norms ||x||p = (sum |xi|^p)^{1/p} for 1 ≤ p < ∞ and the sup norm ||x||∞ = max_i |xi|; function spaces such as C[a, b] with the sup norm, and Lp spaces with ||f||p = (∫ |f|^p)^{1/p}; sequence spaces like l^p. Each defines a norm on a vector space, producing a corresponding normruimten.

Normruimten support a rich theory. A linear map T between normruimten is bounded (and thus continuous) iff there exists a constant C with ||Tx|| ≤ C ||x|| for all x; the smallest such C is the operator norm ||T||. In finite-dimensional spaces, all norms induce the same topology (and all norms are equivalent); in infinite dimensions, the choice of norm affects convergence, compactness, and the structure of the dual space V*, the set of all bounded linear functionals on V.

Normruimten are central in functional analysis and its applications, providing a framework to measure error, study convergence of algorithms, and analyze stability of systems. They underpin classes such as Banach and Hilbert spaces and are foundational in approximation, numerical analysis, and differential equations.