Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $\lVert y_n \rVert_H^2 = \int_0^T \lVert y_n(t) \rVert^2_{V} \leq C\left(\lVert y_0 \rVert^2_H + \int_0^T \lVert f(t) \rVert^2_{V^*}\right)$ for all $n$. Suppose also $y_n \rightharpoonup y$ in $H$.
How do I show that $\lVert y \rVert_{H}^2 = \int_0^T \lVert y(t) \rVert^2_{V} \leq C\left(\lVert y_0 \rVert^2_H + \int_0^T \lVert f(t) \rVert^2_{V^*}\right)$ also holds?