normarendszer

Normarendszer, literally “norm system” in Hungarian, is a concept used in functional analysis to describe a collection of norms defined on a vector space that is used to measure sizes of vectors and to generate a topology.

Definition and scope: Let V be a vector space over the real or complex numbers. A normarendszer on V is a family P = {||·||_i : i ∈ I} of norms on V. The topology on V is commonly taken to be the locally convex topology generated by the associated seminorms p_i(x) = ||x||_i. When I is finite and the norms are equivalent, the system essentially yields the same single-norm topology. If I is countable and the family separates points, the space can be a Fréchet space when it is complete with respect to the induced topology.

Relation to seminorms: While each member of a normarendszer is a norm, in many contexts the same idea is discussed via a directed family of seminorms. A locally convex space can be defined by such a family, where convergence, boundedness, and continuity are described in terms of these seminorms. In this broader sense, a normarendszer provides a flexible framework beyond a single norm.

Examples: In finite-dimensional spaces like R^n, any finite collection of norms defines the same underlying topology, so the choice of norm is largely equivalent for topological purposes. Common examples include the Euclidean norm, the maximum (sup) norm, and the 1-norm. In infinite dimensions, typical norm systems arise in spaces such as the space of smooth functions C∞(Ω), which is a Fréchet space defined by a countable family of seminorms p_m(f) = sup_{x∈K, |α|≤m} |D^α f(x)| over compact subsets K.

Applications: Normarendszer concepts are central to the study of locally convex spaces, distributions, and partial differential equations, where a single norm is insufficient to capture finer aspects of convergence or smoothness.