Closures of quadratic modules

We consider the problem of determining the closure \(\bar M\) of a quadratic module M in a commutative ℝ-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [28][29] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space \(\mathcal{Y}_M\) consisting of all semiorderings lying over M plays an important role in understanding the closure of M. The result of Schmudgen for preorderings in [29] is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of \(\bar M\) which is valid in many cases.