My very tentative idea of the Riesz's representation theorem is that for all functionals in $V^*$ (dual space), $\phi\in V^*$, there exists a unique vector $v\in V$ such that for any other vector $u \in V$, $\phi(u)=\langle u,v\rangle$.
The definition of double vector space involves a mapping from the dual space functionals to the field, $\phi_v\colon V^* \to \mathbb F$, given by $\phi_v(f) = f(v)$.
So the $\forall u\in V$ part is missing in Riesz's representation, but the connection to the dual space in both definitions, and the unique part (isomorphism(?)) make it difficult to avoid comparisons. The problem is that I'm not getting good results when searching for answers.
For example, there is a hint of it in this post, but I'm not familiar with Hilbert spaces.