fa.functional analysis - Topological dual and the notions of "smaller" and "larger" than...

To answer your final question: Let $Phi supset Psi$. Consider $Phi' subset Phi^*$, the former is the continuous linear functionals on $Phi$, and the latter is the set of all linear functionals on $Phi$. Then the restriction of $Phi'$ on $Psi$ is obviously continuous, so $Phi' subset Psi'subset Psi^*$.

Therefore if you make a space smaller, you makes its dual bigger.

Intuitively speaking, elements of $Psi'$ need to be continuous on fewer objects, and hence has fewer constraints; thus $Psi'$ contains more objects.

For your original question: your interpretation is sort-of okay. The point is that infinite dimensional Hilbert spaces admit dense proper subspaces (hope I am getting the notation correct). And in particular you can have two dense subspaces of a Hilbert space with one strictly contained in the other. You may want to review volume 2 of Reed and Simon.