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 < ∞ for any y ∈ Y . Connections of this space with `weak(X) are presented. A sequence x = (xn)n ⊆ X is called B-unconditionally summable if ∑∞ n=1 |〈B(xn, y), z∗〉| < ∞ Preprint submitted to Elsevier 21 December 2007 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.