A norm is a function ‖·‖ from a vector space to [0,∞) assigning each vector a nonnegative length and satisfying positive definiteness (‖v‖=0 iff v=0), absolute homogeneity (‖αv‖=|α|‖v‖), and the triangle inequality (‖u+v‖≤‖u‖+‖v‖).

Principle

The central organizing idea is measurement of vector size compatible with linear structure: homogeneity links scalar multiplication to scale, while the triangle inequality encodes subadditive geometry used to define metrics and topologies on vector spaces.

Demonstration

Standard examples: the Euclidean norm on R^n, p-norms ‖x‖_p=(∑|x_i|^p)^{1/p} for 1≤p<∞, and the supremum norm on C([a,b]). Norms induce metrics d(u,v)=‖u-v‖ and generate the usual notions of convergence and continuity.

Misapplication

Mistakenly assuming any positive functional is a norm without checking positive definiteness (for instance, seminorms may vanish on nonzero vectors). Or using a quasinorm (violating triangle inequality) as if it were a norm in proofs reliant on the triangle inequality.

Consequence

A norm gives a topology and metric structure, enabling analysis of convergence, completeness, compactness (in finite dimensions), and boundedness. Equivalence of norms in finite-dimensional spaces means many analytic properties are independent of the particular norm chosen.

Reversal

The contrast is with seminorms or quasinorms: seminorms satisfy homogeneity and subadditivity but may have nontrivial kernel; quasinorms relax the triangle inequality. Metrics without linear homogeneity also contrast.

Boundary

Norms are defined on vector spaces; they presuppose linear structure and do not encompass arbitrary distance functions. In infinite-dimensional spaces, norms are not all equivalent and topological consequences depend on the chosen norm.

Semantic Tension

Tension exists between norms and related concepts: seminorms, inner-product-induced norms, and gauges or Minkowski functionals. One must distinguish whether the positive definiteness or strict triangle inequality assumptions hold for intended applications.

Synthesis

A norm is the device that measures vector magnitude in a way that harmonizes with scalar multiplication and addition, producing the metric and topological framework used throughout linear analysis.