Seminorm

Summary

A seminorm on a real or complex vector space has all the properties of a norm, except that it may vanish on nonzero vectors. Seminorms are the building blocks of locally convex spaces: a family of seminorms defines a topology, and every locally convex topology arises in this way. Many function spaces are described by seminorms rather than by one norm. An example is the space of continuous functions on $\R$ with uniform convergence on compact sets.