The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval

Let $T$ be a real number with $T>0$ and let $d$ be a natural number. Adopt the notation $\mathcal{B}_{[0,T]}$ and $\lambda_{[0,T]}$ of the restricted Lebesgue measure space on a compact interval, so that $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$ is a measure space. Write $x\cdot y$ for the dot product and $|x|$ for the Euclidean norm of elements $x,y$ of Euclidean space $\mathbb{R}^{d}$.

1. (Square-integrable maps.) A map $u:[0,T]\to\mathbb{R}^{d}$ with components $u^{1},\dots,u^{d}$ is square-integrable if each component $u^{i}$ is measurable from $([0,T],\mathcal{B}_{[0,T]})$ to $\mathbb{R}$ equipped with its Borel $\sigma$-algebra, and the Lebesgue integral of the nonnegative function $t\mapsto|u(t)|^{2}$ is finite:

$$\int_{[0,T]}|u|^{2}\,d\lambda_{[0,T]}<\infty .$$

Write $\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ for the set of all square-integrable maps $[0,T]\to\mathbb{R}^{d}$, equipped with the pointwise operations $(u+v)(t)=u(t)+v(t)$ and $(cu)(t)=c\,u(t)$ for real $c$.

2. (Almost-everywhere equality.) For $u,v\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ write $u\sim v$ if there is a set $N\in\mathcal{B}_{[0,T]}$ with $\lambda_{[0,T]}(N)=0$ such that $u(t)=v(t)$ for every $t\in[0,T]\setminus N$.

3. (The space $L^{2}$.) $L^{2}([0,T];\mathbb{R}^{d})$ is the set of equivalence classes of the relation $\sim$ on $\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$. The class of $u$ is written $[u]$, and the operations on $L^{2}([0,T];\mathbb{R}^{d})$ are $[u]+[v]=[u+v]$ and $c[u]=[cu]$ for real $c$.

4. (Pairing, norm, and metric.) For $u,v\in\mathcal{L}^{2}([0,T];\mathbb{R}^{d})$ set

$$\langle[u],[v]\rangle_{L^{2}}=\int_{[0,T]}u\cdot v\,d\lambda_{[0,T]},\qquad \lVert[u]\rVert_{L^{2}}=\Bigl(\int_{[0,T]}|u|^{2}\,d\lambda_{[0,T]}\Bigr)^{1/2},$$

the second being formed with the nonnegative square root, and

$$d_{L^{2}}([u],[v])=\lVert[u]-[v]\rVert_{L^{2}} .$$

5. (Notational convention.) An element of $L^{2}([0,T];\mathbb{R}^{d})$ is denoted by the same symbol as a representative of it: one writes $u$ for $[u]$, $\langle u,v\rangle_{L^{2}}$ for $\langle[u],[v]\rangle_{L^{2}}$, and $\lVert u\rVert_{L^{2}}$ for $\lVert[u]\rVert_{L^{2}}$.