perusnormin

Perusnormin is a term found in some theoretical discussions to denote a class of basis-aware, norm-like functionals used to measure the size or basic structure of elements in a vector space. It is not part of standard mathematical nomenclature, and its precise definition varies among authors and contexts.

Definition and construction: Let V be a real or complex vector space with a fixed finite basis B = {b1, ..., bn}. Any x ∈ V has coordinates xB = (x1, ..., xn) relative to B, such that x = sum xi bi. A perusnormin associated with B is a function NB: V → R+ defined by NB(x) = f(xB), where f: Rn → R+ is a coordinate function. If f is a norm on Rn, then NB is a norm on V; if f is not a norm, NB may still serve as a size measure or heuristic in analyses that depend on the chosen basis.

Variants and examples: Common choices for f include the l1, l2, or l∞ norms on the coordinate vector xB, yielding NB(x) = ||xB||p for p ∈ {1, 2, ∞}. More generally, f can be a weighted norm, a sum of coordinate functions, or any norm-like function on Rn. Different bases B yield different NB, even for the same vector x.

Usage and reception: Perusnormin is used mainly in discussions of basis-dependent representations, feature scaling, or algorithmic heuristics that exploit a particular coordinate system. Because the concept is not standardized, definitions, properties, and recommended applications vary, and careful specification is required in any formal work.

See also: norm, pseudo-norm, Minkowski functional, basis, coordinate representation. Note: no widely cited formal references establish perusnormin as a canonical mathematical object.