A bilinear version of Orlicz–Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map B: X × Y → Z, a sequence x = (xn)n ⊆ X is called B-absolutely summable if ∞n=1 B(xn, y) Z is finite for any y ∈ Y . Connections of this space with 1 weak(X) are presented. A sequence x = (xn)n ⊆ X is called B-unconditionally summable if ∞n=1 | B(xn, y), z∗ | 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.