seminorm

A seminorm on a vector space V over the real or complex numbers is a function p: V → [0, ∞) that satisfies two basic properties: subadditivity p(x + y) ≤ p(x) + p(y) for all x, y in V, and absolute homogeneity p(αx) = |α| p(x) for all scalars α and all x in V. It is required to be nonnegative.

What distinguishes a seminorm from a norm is definiteness. A norm satisfies p(x) = 0 only when x = 0, while a seminorm may have nonzero vectors with p(x) = 0. The set Ker p = { x in V : p(x) = 0 } is a subspace of V, and one can form the quotient space V / Ker p. On this quotient, the rule ||x + Ker p|| = p(x) defines a norm, making the quotient a normed space.

Seminorms induce a natural pseudometric d(x, y) = p(x − y). This is a metric precisely when p is a norm. Families of seminorms are also central to topology: a locally convex topological vector space is one whose topology can be generated by a family of seminorms.

Examples include: p_f(x) = |f(x)| for a fixed linear functional f, which is a seminorm with Ker p_f = Ker f; p_U(x) = sup{ |f(x)| : f ∈ U } for a set U of linear functionals; and p_K(x) = sup{ |x(t)| : t ∈ K } on spaces of functions, defined by a family of linear functionals. The zero seminorm p ≡ 0 is a simple degenerate example.

Seminorms provide a flexible framework for measuring size while allowing nontrivial kernels, playing a fundamental role in functional analysis and the study of locally convex spaces.