A function ||·|| from a vector space V over R or C to the nonnegative reals satisfying (1) ||v|| ≥ 0 with ||v|| = 0 iff v = 0 (positive definiteness), (2) ||αv|| = |α|·||v|| for scalars α (absolute homogeneity), and (3) ||v+w|| ≤ ||v|| + ||w|| (triangle inequality).

Principle

A norm quantifies vector magnitude in a way compatible with linear structure; it induces a metric d(v,w)=||v−w|| and a topology, and on finite-dimensional spaces any two norms are equivalent.

Demonstration

Euclidean norm on R^n: ||x||_2 = sqrt(x_1^2 + ... + x_n^2) satisfies all axioms; sup norm on C([0,1]) given by ||f||_∞ = sup_{t∈[0,1]} |f(t)| is another example.

Misapplication

Calling a function a norm when it fails positive definiteness (e.g. a seminorm that vanishes on nonzero vectors) or when homogeneity holds only up to a scalar factor without absolute value.

Consequence

Provides notions of convergence, continuity and boundedness; enables use of analytic tools (completeness, Banach space theory) when paired with linear structure and completeness.

Reversal

The inverse notion is a seminorm or pseudonorm where positive definiteness is dropped; such objects measure size up to a kernel of indistinguishable vectors.

Boundary

Defined on vector spaces over R or C in the usual setting; generalizations (norms over modules, extended-value norms, quasi-norms) relax one or more axioms and are outside the strict definition.

Semantic Tension

Tension between 'norm' and 'metric': a metric can exist without coming from a norm (no linear compatibility), and between 'norm' and 'seminorm' where the latter lacks definiteness.

Synthesis

A norm is the linear-algebraic measurement of vector length that obeys homogeneity and the triangle inequality, thereby inducing metric and topological structure essential to analysis and finite- and infinite-dimensional geometry.