Locally convex topological vector space

A Locally convex topological vector space (or Locally convex space) is a topological vector space with the following local convexity condition: there exists a base of neighbourhoods of 0 consisting of convex sets. Equivalently, the topology is that defined by a family of semi-norms. Although they need not be Hausdorff, it is often assumed that they are.

Every Banach space is a locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of Banach spaces. Indeed, local convexity is a generalisation of normable strong enough for the Hahn-Banach theorem to hold, giving a sufficiently rich theory of continous linear functionals.

The examples in the topological vector space article are all locally convex spaces.