When can the natural pairing between a vector space and its dual be thought of as the $L^2$ inner product? I.e. when is there a bijective correspondence between the two?
$\textbf{Edit:}$ Let $V$ be a vector space with dual $V^*$. There's a map $V^* \times V \rightarrow \mathbb{R}$ sending $(\phi,x) \mapsto \phi(x)$. It is often denoted $\phi(x)=\langle\phi,x\rangle$ and referred to as the natural pairing (see section titled "Algebraic Dual Space" at https://en.wikipedia.org/wiki/Dual_space). In certain contexts, I've seen it said that $\langle\phi,x\rangle=\langle\phi,x\rangle_{L^2}.$ For example, if $V=C^\infty(\mathbb{R}),$ then for any fixed $f \in V$, surely $\langle f,-\rangle_{L^2} \in V^*$. So it's not surprising that in certain circumstances, every linear functional on $V$ is of the form $\langle f,-\rangle_{L^2}$. I'm trying to understand when this is the case.
