seminorms

A seminorm on a vector space V over the real or complex field is a function p: V → [0, ∞) that satisfies two basic properties: homogeneity p(αx) = |α| p(x) for all scalars α and all x in V, and subadditivity p(x + y) ≤ p(x) + p(y) for all x, y in V. Unlike a norm, a seminorm need not distinguish all vectors from the zero vector, so it can assign zero to nonzero elements.

The kernel of a seminorm, ker(p) = {x ∈ V : p(x) = 0}, is a subspace of V. The quotient space V/ker(p) can be turned into a normed space by defining ||[x]|| = p(x); this norm is well defined on equivalence classes and captures the seminorm’s identification of vectors that it collapses to zero.

Examples illustrate the distinction between seminorms and norms. The function p(x) = |x1| on R^n is a seminorm that is not a norm since p(x) = 0 for all vectors with first coordinate zero. The function p(x) = max_i |xi| is a norm. In analysis, the Lp seminorms p(f) = (∫ |f|^p)1/p are seminorms on spaces of measurable functions; on appropriate quotient spaces they become norms.

Constructions and contexts: The Minkowski functional (gauge) of a balanced, convex, absorbing set C in V, defined by p(x) = inf{ t > 0 : x ∈ tC }, is a seminorm. A single seminorm or a family of seminorms can generate topologies; in locally convex spaces, the topology is typically defined by a family of seminorms, and continuity of linear maps can be expressed in terms of these seminorms.

In summary, seminorms generalize norms by relaxing the positivity condition, retaining subadditivity and homogeneity, and they play a central role in the theory of locally convex spaces and various functional-analytic constructions.