In this paper, we introduce the quasi 0-gliding hump property of sequence spaces and study a series of elementary properties of spaces of λ-multiplier convergent series. Introduction.Let (X, T ) be a Hausdorff locally convex space, X * the topological dual space of (X, T ) and λ a scalar-valued sequence space. A series j x j in X is said to be λ-multiplier T -convergent if, for each (t j ) ∈ λ, there exists an x ∈ X such that the seriesLet c 00 be the scalar valued sequence space which are 0 eventually, the β-dual space of λ to be defined by: λ β = {(u j ) : j u j t j is convergence for each (t j ) ∈ λ}. It is obvious that if c 00 ⊆ λ, then [λ, λ β ] is a dual pair with respect to the bilinear pairing [ t, ū] = j u j t j , where t = (t j ) ∈ λ, ū = (u j ) ∈ λ β . Let τ (λ, λ β ) denote the Mackey topology of λ with respect to the dual pair [λ, λ β ], i.e., the topology of uniform convergent on all absolutely convex σ(λ β , λ)-compact subsets of λ β , and k(λ, λ β ) the topology of uniform convergent on all σ(λ β , λ)-compact subsets of λ β . It is clear that k(λ, λ β ) is stronger than τ (λ, λ β ).Lemma 1 [14]. Let c 00 ⊆ λ and τ 1 be a vector topology on λ β such that τ 1 is stronger than the coordinate convergence topology. Then the following states are equivalent:(1) B ⊆ λ β is τ 1 -compact;(2) B ⊆ λ β is τ 1 -sequentially compact.

We characterize a nonlinear full invariant of compact Banach-space maps: Let (X, . ) and (Y, . ) be two Banach spaces and P C (X, Y ) be all compact maps which map (X, . ) to (Y, . ). Then each weak operator-topology subseries-convergent series i P i in P c (X, Y ) is also uniform-topology subseries-convergent iff each bounded map from (X, . ) to (l 1 , . 1 ) is a compact map. The necessary condition for each weak operator-topology subseries-convergent series i P i in P C (X, Y ) to be also uniformtopology subseries-convergent is that (X, . ) and (X , . ) both contain no copy of c 0 . This necessary condition is not sufficient.

In this paper, we establish two general Orlicz-Pettis theorems for Gvalued duality pairs, where G is an Abelian topological group. These results give substantial improvements of many important results such as the Vitali-Hahn-Saks theorem, the Hahn-Schur theorem and the Stiles theorem.