$g_n \rightharpoonup g \implies T_{\phi}(g_n) \to T_{\phi} (g)$ ($T_\phi$ is bounded functional)

Question

Put $\dot{H}^1= \{f: \nabla f \in L^2\}.$

Choose $\phi \in \mathcal{S}(\mathbb R^d)$ (Schwartz space), and define $T_{\phi}:\dot{H}^1(\mathbb R^d) \to \mathbb C$ as $T_\phi (f)= \int_{\mathbb R^d} f(x) \phi (x) dx$, so that $T_\phi$ is a bounded linear function on $ \dot{H}^1(\mathbb R^d),$ (that is, $|T_{\phi}(f)| \leq C \|\nabla f\|_{L^2},$ where $C$ is constant)

Assume that $g_n \rightharpoonup g$ in $\dot{H}^1$, that is $\int \nabla g_n \cdot \nabla h \to \int \nabla g \cdot \nabla h$ for all $h\in \dot{H}^1.$

Question: Can we expect that $T_{\phi}(g_n) \to T_{\phi} (g)$ in $\mathbb C$?

Answer 1

Yes, you can use the Riesz representation theorem to get $h \in \dot H^1$ with $$T_\phi(g) = \int \nabla g \cdot \nabla h$$ for all $g \in \dot H^1$.