Banach Algebra of Bounded Functionals

Let V be a non empty additive loop structure and let V1 be a subset of V . We say that V1 has inverse if and only if: (Def. 1) For every element v of V such that v ∈ V1 holds −v ∈ V1. Let V be a non empty additive loop structure and let V1 be a subset of V . We say that V1 is additively-closed if and only if: (Def. 2) V1 is add closed and has inverse. Let V be a non empty additive loop structure. One can verify that ΩV is add closed and has inverse. Let V be a non empty double loop structure. One can verify that every subset of V which is additively-closed is also add closed and has inverse and every subset of V which is add closed and has inverse is also additively-closed.