It is well knon that if $s, t\in\mathbb{R}$, with $s >t$, the following continuous embedding holds $$ H^s(\mathbb{R}^n)\hookrightarrow H^t(\mathbb{R}^n).$$
My question is: these spaces are also dense in each other?
If it is true could anyone give some references?
Moreover, if $H_1$ and $H_2$ are Hilbert spaces and $H_1(\mathbb{R}^n)\hookrightarrow H_2(\mathbb{R}^n)$ is also dense, it is true that also the embedding $$H_2(\mathbb{R}^n)\hookrightarrow H_3(\mathbb{R}^n)$$ is dense, where $H_3(\mathbb{R}^n)$ denotes the dual space of $H_1(\mathbb{R}^n)$?
Could anyone please help? Thank You in advance!
For the first question, you can just use the fact that $C_c^{\infty}(\mathbb{R}^n)$ is a dense subspace of $H^t(\mathbb{R}^n)$ for all $t\in \mathbb{R}$ (see Adams' book 'Sobolev Spaces', theorem 7.38).
For the second question, yes it is true for Hilbert spaces as long as the embedding $H_1\hookrightarrow H_2$ is continuous. More generally on reflexive spaces, you have the following property: $$S\subset H^* \text{ is dense }\iff \left[ x\in H:f(x)=0\;\forall f\in S \implies x=0\right]$$ So if $H_1\hookrightarrow H_2$ is dense and continuous, the embedding $H_2\hookrightarrow H_1^*$ given by $$f(x):=\left \langle f, x\right \rangle_{H_2}\qquad \forall x\in H_1 $$ is such that $$f(x)=0\;\forall f\in H_2\implies \left\langle f,x\right\rangle_{H_2}=0\;\forall f\in H_2 \implies x\in H_2^{\perp}=\left\{0\right\}\implies x=0. $$ So by the above property of reflexive spaces, $H_2$ is dense in $H_1^*$. The embedding is also continuous.