A bilinear Orlicz–Pettis Theorem

Suppose E , F , G are locally convex spaces, b : E × F → G is a bilinear operator and λ is a scalar sequence space. A series ∑ j x j in E is λ b multiplier convergent if for every t = { t j } ∈ λ there exists x t ∈ E such that ∑ j = 1 ∞ t j b ( x j , y ) = b ( x t , y ) 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.