contunuous embedding $\mathcal{W}\subset C(0,T,V)$ and weak convergence question

Question

Consider an evolution triple $(V,H,V^*)$. Let $$\mathcal{V}=L^2(0,T;V),$$ $$\mathcal{W}=\{v\in \mathcal{V}\,|\,v'\in\mathcal{V}^*\}.$$ The embedding $\mathcal{W}\subset C(0,T,H)$ is continuous and we know that $$u_n\to u^*\quad \text{weakly in}\quad \mathcal{V},$$ $$u'_n\to u'^*\quad \text{weakly in}\quad \mathcal{W}.$$

How to deduce (propably applying continuous embedding) that

$$u_n(T)\to u^*(T)\quad \text{weakly in}\quad H,$$ $$u'_n(T)\to u'^*(T)\quad \text{weakly in}\quad H?$$

Answer 1

Define $P\colon \mathcal{V}\to H$ by

$$Pu=u(T).$$ $P$ is bounded and linear opeartor. For any $f\in H^*$, $f\circ P\in \mathcal{V}^*$. Since $(u_m)$ is weakly convergent, then $$(f\circ P)(u_m-u^*)\to 0,$$ implying that $$f(u_m(T)-u^*(T))\to 0.$$