© 1996, Topology Atlas

Nuclear groups were introduced in the monograph Additive subgroups of topological vector spaces, (Springer) Lecture Notes in Math., vol. 1466, in order to clear up various results concerning continuous characters of non-locally compact abelian topological groups. They form a variety of abelian topological groups containing locally compact abelian (LCA) groups and nuclear locally convex spaces. They satisfy the theorems of Bochner (Minlos) and Levy, the Pontryagin duality theorem (under some additional assumptions) and the Glicksberg theorem on weakly compact subsets. From the point of view of convergent series, the properties of nuclear groups are similar to those of nuclear spaces. For instance, unconditionally convergent series are absolutely convergent. The theory of nuclear groups is a geometrical one. It is based on inequalities in the geometry of numbers and in the convex and discrete geometry concerning $n$-dimensional lattices and convex bodies. It also leads to interesting problems concerning linear operators and probability distributions on Banach spaces.