Locally convex space

Summary

A locally convex space is a topological vector space in which every neighbourhood of $0$ contains a convex neighbourhood of $0$. Equivalently, its topology is generated by a family of seminorms. Normed spaces, Fréchet spaces, spaces of smooth functions and spaces of distributions are all locally convex. Local convexity is what makes the Hahn–Banach theorem work, so these spaces have many continuous linear forms and a rich duality theory.