Consider the spaces $H^{1/2}(\mathbb{R}^3), H^{-1/2}(\mathbb{R}^3)$ and $L^{2}(\mathbb{R}^3)$. Are the continuous embeddings true $$H^{1/2}(\mathbb{R}^3)\hookrightarrow L^2(\mathbb{R}^3)\hookrightarrow H^{-1/2}(\mathbb{R}^3)?$$ If the continuous embeddings are true, are they also dense?
Could anyone help, please? Thank you in advance!
As intimated in the comments, use the definition $$H^s(\mathbb{R}^n)=\{u\in\mathcal{S}': \langle\xi\rangle^s\hat{u}\in L^2\}.$$ From here, it's immediately clear that $H^s\subset H^t$ for $s>t,$ owing to the fact that $\langle\xi\rangle^s\geq\langle\xi\rangle^t$ for $s>t.$ In particular, you get your sequence of inclusions.