  A polynomial optimization problem (POP) consists of minimizing a multivariate
real polynomial on a semi-algebraic set $K$ described by polynomial
inequalities and equations. In its full generality it is a non-convex,
multi-extremal, difficult global optimization problem. More than an decade ago,
J.~B.~Lasserre proposed to solve POPs by a hierarchy of convex semidefinite
programming (SDP) relaxations of increasing size. Each problem in the hierarchy
has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP
formulation (a sum-of-squares representation of a polynomial Lagrangian of the
POP). In this note, when the POP feasibility set $K$ is compact, we show that
there is no duality gap between each primal and dual SDP problem in Lasserre's
hierarchy, provided a redundant ball constraint is added to the description of
set $K$. Our proof uses elementary results on SDP duality, and it does not
assume that $K$ has an interior point.
