Weak Convergence in the Lebesgue Space of Square-Integrable Vector-Valued Functions

Let $T>0$ and $d$ be as in the definition of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$, and let $\langle\cdot,\cdot\rangle_{L^{2}}$ be the pairing of that definition. Let $(u_{n})_{n\in\mathbb{N}}$ be a sequence in $L^{2}([0,T];\mathbb{R}^{d})$ and let $u\in L^{2}([0,T];\mathbb{R}^{d})$.

The sequence $(u_{n})_{n\in\mathbb{N}}$ converges weakly to $u$, written $u_{n}\rightharpoonup u$, if for every $v\in L^{2}([0,T];\mathbb{R}^{d})$ the real sequence $\bigl(\langle u_{n},v\rangle_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $\langle u,v\rangle_{L^{2}}$. In that case $u$ is called a weak limit of the sequence.