The $\mathbf{K}$-moment problem for continuous linear functionals

Given a closed (and not necessarily compact) basic semi-algebraic set $\mathbf {K}\subseteq \mathbb{R}^n$, we solve the $\mathbf {K}$-moment problem for continuous linear functionals. Namely, we introduce a weighted $\ell _1$-norm $\ell _{\mathbf {w}}$ on $\mathbb{R}[\mathbf {x}]$, and show that the $\ell _{\mathbf {w}}$-closures of the preordering $P$ and quadratic module $Q$ (associated with the generators of $\mathbf {K}$) is the cone $\textrm {Psd}(\mathbf {K})$ of polynomials nonnegative on $\mathbf {K}$. We also prove that $P$ and $Q$ solve the $\mathbf {K}$-moment problem for $\ell _{\mathbf {w}}$-continuous linear functionals and completely characterize those $\ell _{\mathbf {w}}$-continuous linear functionals nonnegative on $P$ and $Q$ (hence on $\textrm {Psd}(\mathbf {K})$). When $\mathbf {K}$ has a nonempty interior, we also provide in explicit form a canonical $\ell _{\mathbf {w}}$-projection $g^{\mathbf {w}}_f$ for any polynomial $f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $g^{\mathbf {w}}_f$ is very sparse and does not depend on $\mathbf {K}$! This enables us to provide an explicit Positivstellensatz on $\mathbf {K}$. And last but not least, we provide a simple characterization of polynomials nonnegative on $\mathbf {K}$, which is crucial in proving the above results.