standardmetrik

Standardmetrik is a term used in mathematics to denote the canonical metric most naturally associated with a given space. On Euclidean space R^n, the standard metric, often called the Euclidean metric, is defined by d(x,y) = sqrt(sum_{i=1}^n (x_i - y_i)^2). It arises from the standard inner product <x,y> = sum x_i y_i and provides a natural notion of distance between points.

This metric induces the standard topology on R^n, where open sets can be described as unions of open balls. R^n equipped with the standard metric is a complete metric space; geodesics with respect to this metric are straight-line segments. The standard metric thus serves as a fundamental reference in analysis and geometry.

Beyond Euclidean space, the term can refer more generally to the metric generated by a natural norm on a space. For any norm ||·||, the induced distance d(x,y) = ||x - y|| is a standard metric. In finite-dimensional spaces, all norms generate the same topology, so the standard metric is one among several natural choices. Common alternatives include the Manhattan metric d1(x,y) = sum |x_i - y_i| and the sup metric d∞(x,y) = max_i |x_i - y_i|.

Applications of the standard metric span convergence, continuity, and compactness in analysis, geometry, and numerical methods. It also provides baseline distance measures in applied fields such as computer science and data analysis. See also metric, norm, inner product, and topology.