The K-moment Problem for Continuous Linear Functionals

Given a closed (and non necessarily compact) basic semialgebraic set K ⊆ R, we solve the K-moment problem for continuous linear functionals. Namely, we introduce a weighted l1-norm lw on R[x], and show that the lw-closures of the preordering P and quadratic module Q (associated with the generators of K) is the cone Psd(K) of polynomials nonnegative on K. We also prove that P and Q solve the K-moment problem for lw-continuous linear functionals and completely characterize those lw-continuous linear functionals nonnegative on P and Q (hence on Psd(K)). When K has a nonempty interior we also provide in explicit form a canonical lw-projection g w f for any polynomial f , on the (degree-truncated) preordering or quadratic module. Remarkably, the support of gw f is very sparse and does not depend on K! This enables us to provide an explicit Positivstellensatz on K. At last but not least, we provide a simple characterization of polynomials nonnegative on K, which is crucial in proving the above results.