Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.
Real or complex vector spaces with a topology determined by a system of seminorms are called locally convex spaces: alternatively, the topology can be characterized by the feature that $0$ has a system of convex open neighbourhoods. Hilbert spaces, Banach spaces and Frechet spaces are particular examples, but there are more than those: an archetypical example is the space of test functions in distribution theory, a locally convex space which is complete but not Frechet. Questions with this tag concern about topological vector spaces beyond the normed spaces.
