The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis

In the settings of The Real Numbers: Standing Notation and Background and Measure Spaces and the Lebesgue Integral: Standing Notation, the measure space of the latter being instantiated at each use by the measure space named, let $(S,\mathcal{S},\mu)$ be a measure space, and let $E$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{E}$, norm $|\cdot|_{E}$ and an orthonormal basis $(f_{k})_{k\in\mathbb{N}}$. Measurability of maps $S\to E$, their pointwise sums and real multiples, the functions $|v|_{E}^{2}$ and $\langle v,w\rangle_{E}$, and the relation $\sim_{\mu}$ are those of Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality for this $E$, this basis and $(S,\mathcal{S})$.

1. (Square-integrable maps)¶ A measurable map $v:S\to E$ is square-integrable with respect to $\mu$ if $\int_{S}|v|_{E}^{2}\,d\mu<\infty$, the integrand being nonnegative and measurable by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations.

2. (Classes)¶ For a square-integrable $v:S\to E$, its class $[v]$ is the set of the measurable $v':S\to E$ with $v'\sim_{\mu}v$, that is, its equivalence class for the equivalence relation $\sim_{\mu}$ of Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §almost-everywhere; every member of $[v]$ is square-integrable by that clause, and $[v]=[v']$ exactly when $v\sim_{\mu}v'$. The space of square-integrable maps from $S$ to $E$ is the set $L^{2}(\mu;E)$ of all such classes.

3. (Operations, inner product and norm)¶ For square-integrable $v,w:S\to E$ and $t\in\mathbb{R}$,

$$[v]+[w]=[v+w],\qquad t[v]=[tv],\qquad\bigl\langle[v],[w]\bigr\rangle_{L^{2}(\mu;E)}=\int_{S}\langle v,w\rangle_{E}\,d\mu,\qquad\bigl\lVert[v]\bigr\rVert_{L^{2}(\mu;E)}=\Bigl(\int_{S}|v|_{E}^{2}\,d\mu\Bigr)^{1/2};$$

the maps $v+w$ and $tv$ are square-integrable and $\langle v,w\rangle_{E}$ is integrable with respect to $\mu$ by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §square-integrable, and the four right-hand sides depend only on $[v]$ and $[w]$ by Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §almost-everywhere.

4. (Convention)¶ An element of $L^{2}(\mu;E)$ is denoted by the same symbol as a representative of it.