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

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 $E$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{E}$ and norm $|\cdot|_{E}$, let $(f_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $E$, and let $\mathcal{B}(E)$ be the Borel $\sigma$-algebra of $E$ for the distance of its norm. Let $(S,\mathcal{S})$ be a measurable space. Measurability of a map $S\to E$ is taken with respect to $\mathcal{S}$ and $\mathcal{B}(E)$, and of a function $S\to\mathbb{R}$ with respect to $\mathcal{S}$ and the Borel $\sigma$-algebra of the real line, both in the sense of Measurable Function and Real-Valued Measurable Function. For maps $v,w:S\to E$, $t\in\mathbb{R}$ and $k\in\mathbb{N}$, the maps $v+w$ and $tv$ are taken pointwise, $v_{k}:S\to\mathbb{R}$ is the function $v_{k}(s)=\langle v(s),f_{k}\rangle_{E}$, and $|v|_{E}^{2}$ and $\langle v,w\rangle_{E}$ denote the functions $s\mapsto|v(s)|_{E}^{2}$ and $s\mapsto\langle v(s),w(s)\rangle_{E}$.

1. (Operations)¶ Let $v,w:S\to E$ be measurable and $t\in\mathbb{R}$. Then $v+w$ and $tv$ are measurable, and the functions $v_{k}$ ($k\in\mathbb{N}$), $|v|_{E}^{2}$ and $\langle v,w\rangle_{E}$ are measurable. For every $s\in S$ the series $\sum_{k=1}^{\infty}v_{k}(s)^{2}$ converges, and its sum is $|v(s)|_{E}^{2}$.

2. (Synthesis)¶ Let $(g_{k})_{k\in\mathbb{N}}$ be a sequence of measurable functions $g_{k}:S\to\mathbb{R}$ such that for every $s\in S$ the series $\sum_{k=1}^{\infty}g_{k}(s)^{2}$ converges. Then for every $s\in S$ the series $\sum_{k=1}^{\infty}g_{k}(s)f_{k}$ converges in $E$, and the map $v:S\to E$, $v(s)=\sum_{k=1}^{\infty}g_{k}(s)f_{k}$, is measurable with $v_{k}=g_{k}$ for every $k\in\mathbb{N}$.

3. (Square-integrability)¶ Let $\mu$ be a measure on $\mathcal{S}$, let $v,w:S\to E$ be measurable with $\int_{S}|v|_{E}^{2}\,d\mu<\infty$ and $\int_{S}|w|_{E}^{2}\,d\mu<\infty$, and let $t\in\mathbb{R}$. Then $\int_{S}|v+w|_{E}^{2}\,d\mu<\infty$ and $\int_{S}|tv|_{E}^{2}\,d\mu<\infty$; for every $k\in\mathbb{N}$ the function $v_{k}$ is $2$-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space; the function $\langle v,w\rangle_{E}$ is integrable with respect to $\mu$; and

$$\Bigl|\int_{S}\langle v,w\rangle_{E}\,d\mu\Bigr|\le\Bigl(\int_{S}|v|_{E}^{2}\,d\mu\Bigr)^{1/2}\Bigl(\int_{S}|w|_{E}^{2}\,d\mu\Bigr)^{1/2}.$$

4. (Almost-everywhere equality)¶ Let $\mu$ be a measure on $\mathcal{S}$. For measurable $u,u':S\to E$ the set $\{s\in S:u(s)\ne u'(s)\}$ belongs to $\mathcal{S}$; write $u\sim_{\mu}u'$ if it has $\mu$-measure $0$. On the set of measurable maps $S\to E$ the relation $\sim_{\mu}$ is an equivalence relation. Let $v,v',w,w':S\to E$ be measurable with $v\sim_{\mu}v'$ and $w\sim_{\mu}w'$, and let $t\in\mathbb{R}$. Then $v+w\sim_{\mu}v'+w'$ and $tv\sim_{\mu}tv'$. If moreover $\int_{S}|v|_{E}^{2}\,d\mu<\infty$ and $\int_{S}|w|_{E}^{2}\,d\mu<\infty$, then $\int_{S}|v'|_{E}^{2}\,d\mu<\infty$, $\int_{S}|w'|_{E}^{2}\,d\mu<\infty$,

$$\int_{S}|v'|_{E}^{2}\,d\mu=\int_{S}|v|_{E}^{2}\,d\mu,\qquad\int_{S}\langle v',w'\rangle_{E}\,d\mu=\int_{S}\langle v,w\rangle_{E}\,d\mu,$$

and for every $k\in\mathbb{N}$ the classes $[v_{k}]$ and $[v'_{k}]$ of The Lebesgue Space of Power-Integrable Functions §equivalence coincide.