A seminorm is a function defined on a real or complex vector space X, which is almost a norm, but not quite. p: X → [0, ∞) is a seminorm if
- p respects multiplication by scalars: if α is a scalar (in R or C, whichever is the base field of X), then p(αx) = |α| p(x) for every x ∈ X;
- p obeys the triangle inequality: p(x + y) ≤ p(x) + p(y) for every x, y in X;
- p(0) = 0; but we do not require that p(x) ≠ 0 whenever x ≠ 0; p may kill some elements of X. This is what makes p not necessarily a norm.
Families of
seminorms are the easiest way to construct some
categories of
topological vector spaces, such as
Frechet spaces and more generally
locally convex spaces.