Does dense embedding say anything about the embedding of their dual space?

Question

$H^s(\Omega) \subset L^2\cong L^2 \subset H^{-s}(\Omega)$

The embedding is dense from $H^s$ to $L^2$ which is identified with its dual.

Does this tell us the embedding from $L^2$ to $H^{-s}$ is dense?

Here, $\Omega$ is an bounded, open set.

Answer 1

For any continuous linear operator $T:X\to Y$ between normed (or even locally convex) vector spaces you have:

$T$ has dense range $\Leftrightarrow$ $T^t: Y^*\to X^*$ is injective, and

$T$ is injective $\Leftrightarrow$ $T^t$ has weak$^*$ dense range.

Since you are dealing with Hilbert (hence reflexive) spaces you can replace the weak$^*$ density by density w.r.t the dual norm on $X^*$.