In a topological vector space, duality invariant
is a very important property, some famous
theorems, such as the Mackey-Arens theorem, the
Mackey theorem, the Mazur theorem and the
Orlicz-Pettis theorem, all show some duality
invariants. In this paper we would like to show
an important improvement of the invariant
results, which are related to sequential
evaluation convergence of function series.
Especially, a very general invariant result is
established for an abstract mapping
pair  consisting of a nonempty set
 and  is
bounded , where  is a locally convex space.
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