A bilinear Orlicz–Pettis Theorem

Suppose E,F,G are locally convex spaces, is a bilinear operator and λ is a scalar sequence space. A series ∑jxj in E is λ b multiplier convergent if for every t={tj}∈λ there exists xt∈E such that for every y∈F. Under continuity assumptions on the linear operators b(x,⋅), we establish several versions of the Orlicz–Pettis Theorem for multiplier convergent series. Applications to spaces of continuous linear operators are given.