Positive Polynomials and Sequential Closures of Quadratic Modules

Let S be a basic closed semi-algebraic set in R^n and P the corresponding preordering in R[X_1,...,X_n]. We examine for which polynomials f there exist identities f+\ep q \in P for all \ep>0. These are precisely the elements of the sequential closure of P with respect to the finest locally convex topology. We solve the open problem whether this equals the double dual cone of P, by providing a counterexample. We then prove a theorem that allows to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial f that is nonnegative on S admits such representations, or at least the polynomials from the double dual cone of P do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.