Normierten

Normierten is the past participle of the German verb normieren and is used as an attributive or predicate adjective to indicate that something has been brought into a standard form or scaled according to a defined norm. In mathematical language the term often appears in phrases such as normierte Räume or normierte Größen, where a norm assigns a size or length to objects and induces a corresponding metric.

In mathematics, a normierter Raum is a normed space—a vector space equipped with a norm that satisfies positivity, definiteness, homogeneity and the triangle inequality. A vector may be normiert (normalized) by dividing it by its norm, yielding a unit vector v̂ = v / ||v|| for v ≠ 0. The concept is central to analysis and functional analysis, where many constructions rely on the notion of size rather than sheer direction.

In statistics and probability, normierte Daten refers to data that have been standardized, typically by subtracting a mean and dividing by a standard deviation, resulting in zero mean and unit variance. A normierte Dichte (normierte Wahrscheinlichkeitsdichte) is a density function that integrates to one, reflecting a proper probability distribution. In physics, a normierte Wellenfunktion ψ is one whose squared magnitude integrates to unity, ensuring total probability equals one.

Beyond mathematics and science, normieren also appears in information technology and data management as part of database normalization, which reduces redundancy by organizing fields and tables according to a formal standard. The terms normieren and normalisieren are related but carry nuanced usage: normieren stresses adherence to a defined norm, while normalisieren emphasizes bringing quantities to a common scale.