Closed Convex Subsets of the Lebesgue Space of Square-Integrable Vector-Valued Functions are Weakly Sequentially Closed

Let $T>0$ and $d$ be as in the definition of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$, write $H=L^{2}([0,T];\mathbb{R}^{d})$, and adopt the pairing $\langle\cdot,\cdot\rangle_{L^{2}}$, the norm $\lVert\cdot\rVert_{L^{2}}$ and the metric $d_{L^{2}}$ of that definition.

Let $K$ be a subset of $H$ that is convex and closed for the topology of metric open subsets determined by $d_{L^{2}}$, let $(u_{n})_{n\in\mathbb{N}}$ be a sequence in $K$, and let $u\in H$ be such that $u_{n}\rightharpoonup u$ in the sense of weak convergence.

Then $u\in K$.