Positive polynomials and sequential closures of quadratic modules

Let $\mathcal{S}=\{x\in\mathbb{R}^n\mid f_1(x)\geq 0,\ldots,f_s(x)\geq 0\}$ be a basic closed semi-algebraic set in $\mathbb{R}^n$ and let $\mathrm{PO}(f_1,\ldots,f_s)$ be the corresponding preordering in $\mathbb{R}[X_1,\ldots,X_n]$. We examine for which polynomials $f$ there exist identities

$$f+\varepsilon q\in \mathrm{PO}(f_1,\ldots,f_s)$$     for all $$\varepsilon>0.$$

These are precisely the elements of the sequential closure of $\mathrm{PO}(f_1,\ldots,f_s)$ with respect to the finest locally convex topology. We solve the open problem from Kuhlmann, Marshall, and Schwartz (2002, 2005), whether this equals the double dual cone

$$\mathrm{PO}(f_1,\ldots,f_s)^{\vee\vee},$$

by providing a counterexample. We then prove a theorem that allows us 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 $\mathcal{S}$ admits such representations, or at least the polynomials from $\mathrm{PO}(f_1,\ldots,f_s)^{\vee\vee}$ do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.