Norm (mathematics) - Definition

Definition

Given a vector space V over a subfield F of the complex numbers, a norm on V is a function p: V → R with the following properties:

For all a ∈ F and all u, v ∈ V,

1. p(av) = |a| p(v), (positive homogeneity or positive scalability).
2. p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).
3. If p(v) = 0 then v is the zero vector (separates points).

A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus

p(v) ≥ 0 (positivity).

A seminorm is a norm with the 3rd property (separating points) removed.

Every vector space V with seminorm p(v) induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly well-defined and is given by:

p(W + v) = p(v).

A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).