A bilinear version of Orlicz–Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=(xn)n⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=(xn)n⊆X is called B-unconditionally summable if is finite for any y∈Y and z∗∈Z∗ and for any M⊆N there exists xM∈X for which ∑n∈M〈B(xn,y),z∗〉=〈B(xM,y),z∗〉 for all y∈Y and z∗∈Z∗. A bilinear version of Orlicz–Pettis theorem is given in this setting and some applications are presented.