Pontryagin duality for subgroups and quotients of nuclear spaces
Let G be an abelian topological group and \(G^\wedge\) the dual group, i.e. the group of all continuous characters (homomorphisms into R/Z) of G, endowed with the topology of uniform convergence on precompact sets. A subgroup H of G is said to be dually closed if \((G/H)^\wedge\) separates the points of G/H and dually embedded if each \(\chi\in H^\wedge\) can be extended to some \({\tilde \chi}\in G^\wedge\). We say that G is reflexive if the canonical embedding \(G\to G^{\wedge\wedge}\) is a topological isomorphism. A reflexive group G is called strongly reflexive if the duality between G and \(G^\wedge\) induces dualities between appropriate subgroups and quotients of G and \(G^\wedge\). In other words, G is strongly reflexive if every closed subgroup of G and of \(G^\wedge\) is both dually closed and dually embedded and if all closed subgroups and Hausdorff quotient groups of G and of \(G^\wedge\) are reflexive. The following two results are obtained: 1) If K is an additive subgroup of a nuclear space E, then every equicontinuous subset of \(K^\wedge\) is the canonical image of an equicontinuous subset of \(E^\wedge\). In particular, K is dually embedded. 2) Every nuclear Fréchet space is strongly reflexive as an additive topological group. This can be extended to some wider class of groups, e.g. to countable products of nuclear Fréchet spaces and LCA groups.
topology of uniform convergence on precompact sets
Every nuclear Fréchet space is strongly reflexive as an
countable products of nuclear Fréchet spaces
Every nuclear Fréchet space is strongly reflexive as an additive topological group
scientific article; zbMATH DE number 3953539
scientific article