Metrischer

Metrischer is the attributive form of the German adjective metrisch, meaning related to or defined by a metric. In mathematical contexts the term is used to describe objects, structures, or properties that arise from a metric, a function that measures distance between elements of a set.

The most common usage is in the phrase metrischer Raum, or metric space. A metric space is a pair (X, d) where X is a set and d: X×X → [0, ∞) is a metric satisfying four axioms: d(x, y) = 0 if and only if x = y, symmetry d(x, y) = d(y, x), non-negativity, and the triangle inequality d(x, z) ≤ d(x, y) + d(y, z). Distances measured by d induce a topology on X; many classical properties such as convergence, continuity, completeness, and compactness can be expressed in metrischer terms.

Many mathematical structures naturally carry metrische Strukturen. Normed spaces define a metric via d(x, y) = ||x − y||, and inner product spaces induce metrics through d(x, y) = sqrt⟨x − y, x − y⟩. Examples include the Euclidean space with the standard metric and the discrete metric d(x, y) = 1 for x ≠ y. The concept is also used more broadly in other domains where a notion of distance or measure of difference is essential.

Outside mathematics, metrisch can also describe meter-like or metrical aspects, such as poetic meter. In such uses, metrischer corresponds to the masculine nominative singular form of metrisch.