  In this paper we discuss symmetrically self-dual spaces, which are simply
real vector spaces with a symmetric bilinear form. Certain subsets of the space
will be called q-positive, where q is the quadratic form induced by the
original bilinear form. The notion of q-positivity generalizes the classical
notion of the monotonicity of a subset of a product of a Banach space and its
dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss
concepts generalizing the representations of monotone sets by convex functions,
as well as the number of maximally q-positive extensions of a q-positive set.
We also discuss symmetrically self-dual Banach spaces, in which we add a Banach
space structure, giving new characterizations of maximal q-positivity. The
paper finishes with two new examples.
