Let $\{u_{n}\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\mathbb{R})$. Furthermore, assume $u_{n}\to 0$ in $H^{-1}(\mathbb{R})$.
I want to clarify the definition of convergence in $H^{-1}(\mathbb{R})$.
Does this convergence mean the following statement?
$\forall \phi \in H^{-1}(\mathbb{R}),\, _{H^{-1}}\langle \phi,u_{n}\rangle_{H_{0}^{1}}\to 0$ as $n\to\infty$ where $_{H^{-1}}\langle \,\cdot\,,\,\cdot\,\rangle_{H_{0}^{1}}$ is the dual pairing of $H_{0}^{1}$ and $H^{-1}$.
Moreover, since we know that $H_{0}^{1}(\mathbb{R})$ is a Hilbert Space, can we say that it is equivalent to the notion of weak convergence instead?
Any help is pretty much appreciated! Thank you!