I am trying to understand the Hilbert triple $V \subset H \subset V^*$, where $V$ and $H$ are Hilbert spaces and the star denotes the dual space. Eg: $H^1 \subset L^2 \subset H^{-1}.$
If $V \subset H$, then associating it is clear that $H^* \subset V^*$ since every functional in $H^*$ acts on elements of $H$, and since $V$ is in $H$, acts on elements in $V$. Is this understanding right?
Therefore, by assocating $H$ with $H^*$, we can write $V \subset H \equiv H^* \subset V^*$, which is the Hilbert triple.
Is this correct? I feel like I'm missing something deeper.