On characterized subgroups of Abelian topological groups X and the group of all X-valued null sequences

Let X be an Abelian topological group. A subgroup H of X is characterized if there is a sequence u = {u_n} in the dual group of X such that H = {x 2 X : (u_n, x) → 1}. We reduce the study of characterized subgroups of X to the study of characterized subgroups of compact metrizable Abelian groups. Let c0(X) be the group of all X-valued null sequences and u0 be the uniform topology on c₀(X). If X is compact we prove that c₀(X) is a characterized subgroup of X^N if and only if X ̃ Tⁿ × F, where n ≥ 0 and F is a finite Abelian group. For every compact Abelian group X, the group c₀(X) is a g-closed subgroup of X^N . Some general properties of (c₀(X), u₀) and its dual group are given. In particular, we describe compact subsets of (c₀(X), u₀).