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.

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