The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis

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, let $E$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{E}$, norm $|\cdot|_{E}$ and an orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, and let the space $L^{2}(\mu;E)$ of square-integrable maps, with its operations, inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu;E)}$ and norm $\lVert\cdot\rVert_{L^{2}(\mu;E)}$, be those of that definition, and the coordinates $v_{k}(s)=\langle v(s),f_{k}\rangle_{E}$ of maps $v:S\to E$ as in 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. Let $L^{2}(\mu)=L^{2}(S,\mathcal{S},\mu)$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\mu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product.

1. (A real Hilbert space)¶ With the operations, inner product and norm of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $L^{2}(\mu;E)$ is a real Hilbert space whose zero vector is the class of the constant map with value the zero vector of $E$, and $\lVert\cdot\rVert_{L^{2}(\mu;E)}$ is the norm of its inner product.

2. (Coordinates)¶ For $v\in L^{2}(\mu;E)$ and $k\in\mathbb{N}$, the class in $L^{2}(\mu)$ of the coordinate $v_{k}$ of a representative of $v$ does not depend on the representative; it is again written $v_{k}$. For $v,w\in L^{2}(\mu;E)$, $t\in\mathbb{R}$ and $k\in\mathbb{N}$ one has $(v+w)_{k}=v_{k}+w_{k}$ and $(tv)_{k}=t\,v_{k}$; the series $\sum_{k=1}^{\infty}\langle v_{k},w_{k}\rangle_{L^{2}(\mu)}$ and $\sum_{k=1}^{\infty}\lVert v_{k}\rVert_{L^{2}(\mu)}^{2}$ converge, with

$$\langle v,w\rangle_{L^{2}(\mu;E)}=\sum_{k=1}^{\infty}\langle v_{k},w_{k}\rangle_{L^{2}(\mu)},\qquad\lVert v\rVert_{L^{2}(\mu;E)}^{2}=\sum_{k=1}^{\infty}\lVert v_{k}\rVert_{L^{2}(\mu)}^{2}.$$

3. (Synthesis)¶ Let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in $L^{2}(\mu)$ such that the series $\sum_{k=1}^{\infty}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}$ converges. Then there is exactly one $v\in L^{2}(\mu;E)$ with $v_{k}=g_{k}$ for every $k\in\mathbb{N}$.