So I am confused about something very basic. I'm going to outline my confusion, and would love if someone could point out when I'm saying something wrong.
Let $H$ be a Hilbert space. It's dual $H^*$ can be identified with $H$ by Reisz representation theorem. Now let $A$ be a closed subspace. $A$ is then a Hilbert space in its own right, and so it's dual $A^*$ can be identified with $A$ again by Reisz representation. Therefore we have a canonical injection $i: A^* \rightarrow H^*$.
On the other hand to every element in $H^*$, say $\phi$, we can consider the restriction $\phi|_A$ as a bounded linear operator on $A$, and so $\phi|_A \in A^*$. Therefore, I have created a map $j: H^* \rightarrow A^*$.
In finite dimensions, it is clear $j$ is not injective just by considering $H = R^n, A = R^k, k < n$. So the right way to think about this is that as the space gets higher in dimension the dual space also gets higher in dimension.
However, when dealing with infinite dimensional spaces like $H^s, L2,$ etc... I am used to thinking that as the space shrinks the dual space grows. For instance in PDE we freely use the fact that $H_0^1 \subset L2 \Rightarrow H^{-1} \supset L2^* = L2$ How do I reconcile these two points of view? Is there something strange with regards to considering consistent inner products?