Construction of Banach spaces using a generalization of the scalar product

A vector is assigned to every pair of elements of a linear space. If this vector is in $l_2$ and the relation between the element pair and the vector satisfies a certain system of axioms, then we call the space a Banach space. For such a space we introduce, as in the case of the usual scalar product, a series of concepts, and prove, as an example a theorem concerning orthogonal expansion.