In the Wikipedia page of Hahn-Banach theorem, it mentions, when a normed space $X$ is reflexive, then the following "vector problem" is equivalent to the "functional problem".
(The vector problem) Given a collection $(f_i)_{i \in I}$ of bounded linear functionals on a normed space $X$ and a collection of scalars $\left(c_{i}\right)_{i\in I}$ determine if there is an $x\in X$ such that $f_{i}(x)=c_{i}$ for all $i\in I$.
(The functional problem) Given a collection $\left(x_{i}\right)_{i\in I}$ of vectors in a normed space $X$ and a collection of scalars $\left(c_{i}\right)_{i\in I}$, determine if there is a bounded linear functional $f$ on $X$ such that $f\left(x_{i}\right)=c_{i}$ for all $i\in I$.
My question is, can we see the moment problem as a special case of the "functional problem" above? It seems the difficulty is to find a suitable normed space containing all the univariate monomials $\left\{x^n: n\in\mathbb{N}\right\}$.
(The moment problem) Given a sequence of real numbers $(m_n)_n\in\mathbb{R}$, find a measure $\mu$ such that $$m_n=\int_{-\infty}^{+\infty}x^n d\mu(x).$$