pseudonorm

A pseudonorm is a function p from a vector space V over the real or complex numbers to the nonnegative reals that satisfies p(αx) = |α| p(x) for all scalars α and p(x + y) ≤ p(x) + p(y) for all x, y in V. It is nonnegative and p(0) = 0. Unlike a norm, a pseudonorm is not required to separate points; that is, it may assign zero to nonzero vectors. If p(x) = 0 implies x = 0, then p is a norm.

In relation to seminorms, a pseudonorm often serves as a seminorm: positive homogeneity and subadditivity, plus nonnegativity, but potentially nontrivial kernel. The set {x ∈ V : p(x) = 0} is a linear subspace called the kernel. The quotient space V / ker(p) inherits a norm from p, defined by p̄([x]) = p(x). Pseudonorms (as a family or individually) are used to define locally convex topologies on V, since families of seminorms generate such topologies.

Example: On R^2, the function p(x, y) = |x| is a pseudonorm. It is homogeneous and subadditive, and p(0) = 0, but p(x, y) = 0 for all vectors with x = 0, which are nonzero. This illustrates the lack of definiteness and why p is not a norm, though it is a valid seminorm.

Use and terminology: In many texts, pseudonorm and seminorm are used interchangeably, while others distinguish pseudonorm as a non-definite form and reserve seminorm for the broader concept. Regardless, pseudonorms are fundamental in the study of locally convex spaces, duality, and quotient constructions.