On the Group of Absolutely Summable Sequences

Abstract

For an abelian topological group G, the sequence group ℓ1(G) of all absolutely summable sequences in G is studied. It is shown that ℓ1(G) is a Pontryagin reflexive group in case G is a reflexive metrizable group or an LCA group. Further, ℓ1(G) has the Schur property if and only if G has it and ℓ1(G) is a Schwartz group if and only if G is linearly topologized.

Keywords

• summable sequence
• absolutely summable sequence
• locally quasi-convex group
• Schwartz group
• nuclear group
• Pontryagin reflexive group