Completeness of 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 adopt the notation of that definition together with that of the inner-product lemma for $L^{2}([0,T];\mathbb{R}^{d})$. Then the following hold.

1. (Completeness.) The metric space $\bigl(L^{2}([0,T];\mathbb{R}^{d}),d_{L^{2}}\bigr)$ is complete.

2. (Almost-everywhere convergent subsequence.) Let $u_{n}$ ($n\in\mathbb{N}$) and $u$ belong to $\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ and suppose the real sequence $\bigl(\lVert[u_{n}]-[u]\rVert_{L^{2}}\bigr)_{n\in\mathbb{N}}$ has limit $0$. Then there exist natural numbers $n_{1}<n_{2}<n_{3}<\dots$ and a set $N\in\mathcal{B}_{[0,T]}$ with $\lambda_{[0,T]}(N)=0$ such that for every $t\in[0,T]\setminus N$ the sequence $\bigl(u_{n_{j}}(t)\bigr)_{j\in\mathbb{N}}$ converges to $u(t)$ in $\mathbb{R}^{d}$ with the Euclidean distance.