Separability 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, of the inner-product lemma, and the level-$m$ dyadic atoms $I_{m,p}$ of the dyadic averaging lemma. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$.

Let $\mathbb{Q}^{d}$ denote the set of points of $\mathbb{R}^{d}$ all of whose coordinates are rational, that is, the set of $d$-tuples in $\mathbb{Q}$, a point of $\mathbb{R}^{d}$ being identified throughout with the $d$-tuple of its coordinates. For a natural number $m$ let $D_{m}$ be the set of maps $[0,T]\to\mathbb{R}^{d}$ of the form

$$w=\sum_{p=1}^{2^{m}}c_{p}\,\mathbf{1}_{I_{m,p}},\qquad c_{1},\dots,c_{2^{m}}\in\mathbb{Q}^{d},$$

and put $D=\bigcup_{m\in\mathbb{N}}D_{m}$, where $\mathbb{N}$ is the set of natural numbers. Then the following hold.

1. (A countable family.) Every $w\in D$ lies in $\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$, and $D$ is countable.

2. (Density.) For every $u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ and every real $\varepsilon>0$ there is $w\in D$ with $\lVert[u]-[w]\rVert_{L^{2}}<\varepsilon$.

Consequently the set $\{[w]:w\in D\}$ is a dense subset of the metric space $\bigl(L^{2}([0,T];\mathbb{R}^{d}),d_{L^{2}}\bigr)$ and is itself countable.