Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

We start by an application the of Krein-Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn-Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval [0,+infinity). Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.