Let V=V0: be a vector space over a field F wit11 associated spaces of linear functionals V1, V2, V3, V4, Vjbelonging to Hom (Vj-1,F).In this paper we generalize the procedure used to extend a binary product on V to one on V2,focusing here on the case where V is a triple system. As in the case of binary algebras we have a space [Vbreve] (called the closure of V), a subtriple system of V2whlich satisfies the same multilinear identies as V. 1n paritcular, if V is a Jordan triple system and the characteristic of F is not two or three, then ˘ isalso a Jordan triple system; if V is a Lie triple system, then [Vbreve] is also a Lie triple system.