Normabhängig

Normabhängig describes a quantity, property, or construction in mathematics that depends only on the norm of a vector in a normed space. In other words, its value is determined by the magnitude ||x|| of the vector x, not by its direction. Such quantities are often described as radial or norm-driven because they remain constant on spheres of equal radius.

Characterization and implications: A function f: V -> R is normabhängig if there exists a univariate function h such that f(x) = h(||x||) for all x in V. Then f is invariant under all linear isometries that preserve the norm, such as rotations and reflections in Euclidean space. This makes normabhängige quantities sensitive only to the size of the vector, not to its orientation.

Examples and non-examples: The norms themselves, such as ||x||p, are normabhängig. More generally, any function of the form f(x) = h(||x||) is normabhängig. Distances defined with a fixed norm, for example d(x,S) = inf_{y in S} ||x - y||, are normabhängig as long as the underlying norm is fixed. By contrast, a quantity that depends on direction, such as x1 or any linear functional a·x with a fixed vector a, is not normabhängig because it changes with orientation even for vectors of the same magnitude.

Applications and notes: Normabhängige quantities are central in analysis, optimization, and machine learning because their geometry is governed by the chosen norm. They are particularly relevant for understanding rotational or scale invariance and for comparing problems under different norms. See also norms, normed spaces, p-norms, dual norms, and rotational invariance.