This may be a rather dense question, but I would nevertheless be grateful for some guidance.
The question has to do with the Sobolev spaces $H^m (\Omega)$ on an open bounded domain $\Omega$ of $\mathbb{R}^n$, where $m \geq 0$ is integer. These are Hilbert spaces; thus the Riesz representation theorem tells us that there is a one-to-one and onto correspondence between elements of $H^m (\Omega)$ and elements of its dual $H^{-m} (\Omega)$. Yet, we also have the inclusion property $H^{m} (\Omega) \subset L_2 (\Omega) \subset H^{-m} (\Omega)$ (e.g. Oden and Reddy, Intro. to the Mathematical Theory of Finite Elements, p. 108). How can both of these hold? In other words, how can there be a one-to-one and onto correspondence between $H^m (\Omega)$ and $H^{-m} (\Omega)$, and yet the former is a strict subset of the latter?