Basic Properties of 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 $L^{2}([0,T];\mathbb{R}^{d})$, 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, together with weak convergence in $H$. Let $(u_{n})_{n\in\mathbb{N}}$ and $(v_{n})_{n\in\mathbb{N}}$ be sequences in $H$ and let $u,u',v\in H$. Then the following hold.

1. (Uniqueness of the weak limit.) If $u_{n}\rightharpoonup u$ and $u_{n}\rightharpoonup u'$, then $u=u'$.

2. (Strong convergence implies weak convergence.) If the real sequence $\bigl(\lVert u_{n}-u\rVert_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $0$, then $u_{n}\rightharpoonup u$.

3. (The norm bound passes to the weak limit.) If $u_{n}\rightharpoonup u$ and $C$ is a real number with $\lVert u_{n}\rVert_{L^{2}}\le C$ for every $n\in\mathbb{N}$, then $\lVert u\rVert_{L^{2}}\le C$.

4. (Pairing a strongly convergent with a bounded weakly convergent sequence.) Suppose $u_{n}\rightharpoonup u$, the real sequence $\bigl(\lVert v_{n}-v\rVert_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $0$, and there is a real number $C$ with $\lVert u_{n}\rVert_{L^{2}}\le C$ for every $n$. Then the real sequence $\bigl(\langle u_{n},v_{n}\rangle_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $\langle u,v\rangle_{L^{2}}$.

5. (Testing weak convergence on a dense set.) Suppose there is a real number $C$ with $\lVert u_{n}\rVert_{L^{2}}\le C$ for every $n$, and let $E$ be a dense subset of the metric space $(H,d_{L^{2}})$ such that for every $w\in E$ the real sequence $\bigl(\langle u_{n},w\rangle_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $\langle u,w\rangle_{L^{2}}$. Then $u_{n}\rightharpoonup u$.