Bounded Sequences in the Lebesgue Space of Square-Integrable Vector-Valued Functions Have Weakly Convergent Subsequences

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 $(u_{n})_{n\in\mathbb{N}}$ be a sequence in $H$ and let $C$ be a real number with $\lVert u_{n}\rVert_{L^{2}}\le C$ for every $n\in\mathbb{N}$.

Then there are natural numbers $n_{1}<n_{2}<n_{3}<\dots$ and an element $u\in H$ such that the subsequence $(u_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $u$, and moreover $\lVert u\rVert_{L^{2}}\le C$.