In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. Every normed space is locally convex, since the triangle inequality ensures that all balls are convex.

More formally, 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 such a space need not be Hausdorff, this is often also assumed.

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 continuous linear functionals.

Many examples of locally convex topological vector spaces are described in the topological vector space article. On the other hand, L^p spaces for 0 < p < 1 are not locally convex.