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

Each result cited is universally quantified over the data in its own statement.

The claims are proved in the order 2, 3, 1; the proofs of claims 2 and 3 use only The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis and the operations of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, not claim 1. Write $\mathcal{L}^{2}$ for the set of $2$-integrable functions on $(S,\mathcal{S},\mu)$; $L^{2}(\mu)$ carries the operations of The Lebesgue Space of Power-Integrable Functions §space and is a real inner product space by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product. For measurable $u,u':S\to E$ put $D(u,u')=\{s\in S:u(s)\ne u'(s)\}$, a member of $\mathcal{S}$ 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. We record three preliminary facts.

Preliminaries. (P1) A measurable function $S\to\mathbb{R}$ with nonnegative values is a measurable map into $[0,\infty]$ by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable; if it is integrable, then by Integrable Function and the Lebesgue Integral its integral equals its integral as a nonnegative function, its positive part being itself and its negative part the zero function, whose integral is $0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral; conversely, if its integral as a nonnegative function is finite, it is integrable by the criterion recorded there. (P2) For $f\in\mathcal{L}^{2}$, The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and Real Inner Product Space §norm give $\lVert[f]\rVert_{L^{2}(\mu)}^{2}=\langle[f],[f]\rangle_{L^{2}(\mu)}=\int_{S}f^{2}\,d\mu$, the integral being that of a nonnegative function by (P1). For square-integrable $v:S\to E$, the function $\langle v,v\rangle_{E}=|v|_{E}^{2}$ is integrable 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 by (P1) and Properties of Real Powers of Nonnegative Real Numbers §inverse with Properties of Real Powers of Nonnegative Real Numbers §agreement,

$$\bigl\langle[v],[v]\bigr\rangle_{L^{2}(\mu;E)}=\int_{S}|v|_{E}^{2}\,d\mu=\bigl\lVert[v]\bigr\rVert_{L^{2}(\mu;E)}^{2}.$$

(P3) If $h_{k}:S\to\mathbb{R}$ ($k\in\mathbb{N}$) are measurable with nonnegative values and $H_{n}=\sum_{k=1}^{n}h_{k}$, then $H=\sup_{n}H_{n}:S\to[0,\infty]$ is measurable and $\int_{S}H\,d\mu=\sup_{n}\sum_{k=1}^{n}\int_{S}h_{k}\,d\mu$ in $[0,\infty]$: the $H_{n}$ are measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and nondecreasing in $n$, so Monotone Convergence Theorem gives $\int_{S}H\,d\mu=\sup_{n}\int_{S}H_{n}\,d\mu$, and $\int_{S}H_{n}\,d\mu=\sum_{k=1}^{n}\int_{S}h_{k}\,d\mu$ by induction on $n$ from the additivity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative.

Claim 2. Let $v,v'$ be two representatives of the same element of $L^{2}(\mu;E)$; they are square-integrable with $v\sim_{\mu}v'$ by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes, so $v_{k}$ and $v'_{k}$ lie in $\mathcal{L}^{2}$ 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 $[v_{k}]=[v'_{k}]$ 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. Now let $v,w$ be square-integrable representatives and $t\in\mathbb{R}$. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations the pointwise maps $v+w$ and $tv$ represent the sum and the multiple, and for $s\in S$ conditions (b) and (c) of the definition of an inner product give $(v+w)_{k}(s)=v_{k}(s)+w_{k}(s)$ and $(tv)_{k}(s)=t\,v_{k}(s)$; hence by The Lebesgue Space of Power-Integrable Functions §space the class of $(v+w)_{k}$ is $[v_{k}]+[w_{k}]$ and that of $(tv)_{k}$ is $t[v_{k}]$. For the norm identity, the functions $v_{k}^{2}$ are measurable and nonnegative 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 and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; for each $s$ the series $\sum_{k}v_{k}(s)^{2}$ converges with sum $|v(s)|_{E}^{2}$ 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, which equals $\sup_{n}\sum_{k=1}^{n}v_{k}(s)^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion. By (P3) and (P2), $\int_{S}|v|_{E}^{2}\,d\mu=\sup_{n}\sum_{k=1}^{n}\lVert v_{k}\rVert_{L^{2}(\mu)}^{2}$. The left side is finite, so the partial sums of $\sum_{k}\lVert v_{k}\rVert_{L^{2}(\mu)}^{2}$ are bounded above, and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion and (P2) give that this series converges with sum $\int_{S}|v|_{E}^{2}\,d\mu=\lVert v\rVert_{L^{2}(\mu;E)}^{2}$. For the inner product identity, write $u=v+(-1)w$; the maps $v+w$ and $u$ are square-integrable 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 for every $s$, since $(-1)x=-x$ in $E$ by claim 5 of Elementary Identities in a Vector Space, Elementary Identities in a Real Inner Product Space §polarisation gives $4\langle v(s),w(s)\rangle_{E}=|v(s)+w(s)|_{E}^{2}-|u(s)|_{E}^{2}$. All three functions are integrable (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 (P1)), so by Linearity and Monotonicity of the Lebesgue Integral §integrable, (P1) and the norm identity just proved, applied to $v+w$ and to $u$, whose coordinate classes are $v_{k}+w_{k}$ and $v_{k}+(-1)w_{k}=v_{k}-w_{k}$ by linearity and claim 5 of Elementary Identities in a Vector Space,

$$4\bigl\langle v,w\bigr\rangle_{L^{2}(\mu;E)}=\int_{S}|v+w|_{E}^{2}\,d\mu-\int_{S}|u|_{E}^{2}\,d\mu=\sum_{k=1}^{\infty}\lVert v_{k}+w_{k}\rVert_{L^{2}(\mu)}^{2}-\sum_{k=1}^{\infty}\lVert v_{k}-w_{k}\rVert_{L^{2}(\mu)}^{2}.$$

By Elementary Properties of Series of Real Numbers §linearity, applied first with $\lambda=-1$ to the second series and then to the sum of the first series and the resulting one, the difference of these two convergent series is the convergent series with terms $\lVert v_{k}+w_{k}\rVert_{L^{2}(\mu)}^{2}-\lVert v_{k}-w_{k}\rVert_{L^{2}(\mu)}^{2}$, which equal $4\langle v_{k},w_{k}\rangle_{L^{2}(\mu)}$ by Elementary Identities in a Real Inner Product Space §polarisation in $L^{2}(\mu)$. Applying Elementary Properties of Series of Real Numbers §linearity once more, with $\lambda=\frac{1}{4}$, the series $\sum_{k}\langle v_{k},w_{k}\rangle_{L^{2}(\mu)}$ converges with sum $\frac{1}{4}\cdot4\langle v,w\rangle_{L^{2}(\mu;E)}=\langle v,w\rangle_{L^{2}(\mu;E)}$.

Claim 3. Existence: choose $h_{k}\in\mathcal{L}^{2}$ with $[h_{k}]=g_{k}$, and put $G_{n}=\sum_{k=1}^{n}h_{k}^{2}$ and $G=\sup_{n}G_{n}:S\to[0,\infty]$, the functions $h_{k}^{2}$ being measurable and nonnegative by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By (P3) and (P2), $\int_{S}G\,d\mu=\sup_{n}\sum_{k=1}^{n}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}=\sum_{k=1}^{\infty}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}<\infty$, the last equality by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §finite, $N=\{s\in S:G(s)=\infty\}$ belongs to $\mathcal{S}$ and $\mu(N)=0$. Let $\tilde g_{k}=\mathbf{1}_{S\setminus N}h_{k}$, measurable by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $s\in N$ every $\tilde g_{k}(s)$ is $0$, so $\sum_{k}\tilde g_{k}(s)^{2}$ converges with sum $0$; for $s\notin N$ the partial sums of $\sum_{k}\tilde g_{k}(s)^{2}$ are the numbers $G_{n}(s)\le G(s)<\infty$, so the series converges with sum $G(s)$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion. 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 §synthesis the map $v(s)=\sum_{k}\tilde g_{k}(s)f_{k}$ is measurable with $v_{k}=\tilde g_{k}$, and 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 $|v(s)|_{E}^{2}=\sum_{k}\tilde g_{k}(s)^{2}\le G(s)$ for every $s$. Hence $\int_{S}|v|_{E}^{2}\,d\mu\le\int_{S}G\,d\mu<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, so $v$ is square-integrable and $[v]\in L^{2}(\mu;E)$. Its coordinate $v_{k}=\tilde g_{k}$ lies in $\mathcal{L}^{2}$ 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 agrees with $h_{k}$ off the null set $N$, so $v_{k}=[h_{k}]=g_{k}$ in $L^{2}(\mu)$ by The Lebesgue Space of Power-Integrable Functions §equivalence. Uniqueness: let $v,v'$ be square-integrable with $[v_{k}]=[v'_{k}]=g_{k}$ for every $k$, and put $u=v+(-1)v'$, which represents $[v]+(-1)[v']$ by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations. By claim 2, $u_{k}=g_{k}+(-1)g_{k}$, which is the zero vector of $L^{2}(\mu)$ because $(-1)g_{k}=-g_{k}$ by claim 5 and $g_{k}+(-g_{k})$ is the zero vector by claim 2 of Elementary Identities in a Vector Space, and whose norm is therefore $0$ by Elementary Identities in a Real Inner Product Space §zero, so $\int_{S}|u|_{E}^{2}\,d\mu=\sum_{k}0=0$ by claim 2 and (P2). By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing there is $B\in\mathcal{S}$ with $\mu(B)=0$ containing every $s$ with $|u(s)|_{E}^{2}\ne0$. Since $u(s)=v(s)-v'(s)$ by claim 5 of Elementary Identities in a Vector Space, and for $x,y\in E$ one has $x-y=0_{E}$ exactly when $x=y$ (if $x=y$ this is claim 2 there; conversely, if $x-y=0_{E}$, then by condition 2 of Vector Space over a Field both $w=x$ and $w=y$ satisfy $(-y)+w=0_{E}$, so $x=y$ by the uniqueness in claim 2 there), Elementary Identities in a Real Inner Product Space §vanishing shows that $v(s)\ne v'(s)$ exactly when $|u(s)|_{E}\ne0$, that is, exactly when $|u(s)|_{E}^{2}\ne0$, so $D(v,v')\subseteq B$ and $\mu(D(v,v'))=0$ by Basic Properties of a Measure §monotone. Thus $v\sim_{\mu}v'$ and $[v]=[v']$ by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes.

Claim 1. Vector space: let $z:S\to E$ be the constant map with value $0_{E}$; the preimage under $z$ of any set is $\varnothing$ or $S$, both in $\mathcal{S}$, so $z$ is measurable, and $|z|_{E}^{2}=0$ by Elementary Identities in a Real Inner Product Space §zero, so $\int_{S}|z|_{E}^{2}\,d\mu=0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral and $[z]\in L^{2}(\mu;E)$. For square-integrable $u,v,w$ and $\lambda,\kappa\in\mathbb{R}$, by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations each side of each of the conditions 1, 2, 5, 6, 7, 8 of Vector Space over a Field for $[u],[v],[w],\lambda,\kappa$ is the class of the map whose value at each $s$ is the corresponding side of the same condition in $E$ for $u(s),v(s),w(s),\lambda,\kappa$; these maps coincide since $E$ is a vector space, so the classes coincide. Further $[v]+[z]=[v+z]=[v]$ and $[v]+[(-1)v]=[v+(-1)v]=[z]$, because $v(s)+(-1)v(s)=0_{E}$ by claim 5 of Elementary Identities in a Vector Space. Hence $L^{2}(\mu;E)$ is a real vector space; by claim 1 of Elementary Identities in a Vector Space its zero vector is $[z]$, and by claim 5 there $[x]-[y]=[x]+(-1)[y]$. Inner product: for square-integrable $u,v,w$ and $\lambda\in\mathbb{R}$ the functions $\langle u,w\rangle_{E}$, $\langle v,w\rangle_{E}$, $\langle u+v,w\rangle_{E}$, $\langle\lambda u,w\rangle_{E}$ and $\langle w,u\rangle_{E}$ are integrable 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 pointwise $\langle u(s),w(s)\rangle_{E}=\langle w(s),u(s)\rangle_{E}$, $\langle u(s)+v(s),w(s)\rangle_{E}=\langle u(s),w(s)\rangle_{E}+\langle v(s),w(s)\rangle_{E}$ and $\langle\lambda u(s),w(s)\rangle_{E}=\lambda\langle u(s),w(s)\rangle_{E}$ by (a), (b), (c) of Real Inner Product Space §inner-product; integrating with Linearity and Monotonicity of the Lebesgue Integral §integrable gives (a), (b), (c) for $\langle\cdot,\cdot\rangle_{L^{2}(\mu;E)}$. For (d), (P2) gives $\langle[v],[v]\rangle_{L^{2}(\mu;E)}=\int_{S}|v|_{E}^{2}\,d\mu\ge0$; if it is $0$, then by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing there is $B\in\mathcal{S}$ with $\mu(B)=0$ containing every $s$ with $|v(s)|_{E}^{2}\ne0$, which by Elementary Identities in a Real Inner Product Space §vanishing is the set $D(v,z)$, so $\mu(D(v,z))=0$ by Basic Properties of a Measure §monotone, $v\sim_{\mu}z$ and $[v]=[z]$. Thus $L^{2}(\mu;E)$ is a real inner product space, and since $\lVert[v]\rVert_{L^{2}(\mu;E)}\ge0$ has square $\langle[v],[v]\rangle_{L^{2}(\mu;E)}$ by (P2), it is the norm of Real Inner Product Space §norm; the distance is $d(x,y)=\lVert x-y\rVert_{L^{2}(\mu;E)}$ by Real Inner Product Space §distance. For $x,y\in L^{2}(\mu;E)$, claim 2 and claim 5 of Elementary Identities in a Vector Space give $(x-y)_{k}=x_{k}-y_{k}$, hence by claim 2 and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, for every $K\in\mathbb{N}$,

$$\sum_{k=1}^{K}\lVert x_{k}-y_{k}\rVert_{L^{2}(\mu)}^{2}\le\lVert x-y\rVert_{L^{2}(\mu;E)}^{2},\qquad\lVert x-y\rVert_{L^{2}(\mu;E)}^{2}=\sum_{k=1}^{\infty}\lVert x_{k}-y_{k}\rVert_{L^{2}(\mu)}^{2}.\tag{$*$}$$

Completeness: let $(v^{j})_{j\in\mathbb{N}}$ be a Cauchy sequence in $(L^{2}(\mu;E),d)$. By the first part of $(*)$, for each $k$ the sequence $(v^{j}_{k})_{j\in\mathbb{N}}$ is Cauchy in $L^{2}(\mu)$, which is complete by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert and Real Hilbert Space §hilbert; let $g_{k}\in L^{2}(\mu)$ be its limit. Let $\varepsilon>0$; first choose $J\in\mathbb{N}$ with $d(v^{i},v^{j})<\varepsilon$ for all $i,j\ge J$, then fix $j\ge J$ and $K\in\mathbb{N}$. For every $i\ge J$, $(*)$ gives $\sum_{k=1}^{K}\lVert v^{j}_{k}-v^{i}_{k}\rVert_{L^{2}(\mu)}^{2}<\varepsilon^{2}$. As $i\to\infty$, $v^{j}_{k}-v^{i}_{k}$ converges to $v^{j}_{k}-g_{k}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits and its norm converges to $\lVert v^{j}_{k}-g_{k}\rVert_{L^{2}(\mu)}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, both in $L^{2}(\mu)$; passing to the limit in the finite sum,

$$\sum_{k=1}^{K}\lVert v^{j}_{k}-g_{k}\rVert_{L^{2}(\mu)}^{2}\le\varepsilon^{2}\qquad(j\ge J,\ K\in\mathbb{N}).\tag{$**$}$$

Take $\varepsilon=1$ and the corresponding $J$. For each $k$, $g_{k}=(g_{k}-v^{J}_{k})+v^{J}_{k}$, so The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and Elementary Identities in a Real Inner Product Space §homogeneity in $L^{2}(\mu)$ give $\lVert g_{k}\rVert\le\lVert v^{J}_{k}-g_{k}\rVert+\lVert v^{J}_{k}\rVert$, whence $\lVert g_{k}\rVert^{2}\le2\lVert v^{J}_{k}-g_{k}\rVert^{2}+2\lVert v^{J}_{k}\rVert^{2}$ (all norms in $L^{2}(\mu)$). Summing over $k\le K$ and using $(**)$ and claim 2 with Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, $\sum_{k=1}^{K}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}\le2+2\lVert v^{J}\rVert_{L^{2}(\mu;E)}^{2}$ for every $K$, so $\sum_{k}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion. By claim 3 there is $v\in L^{2}(\mu;E)$ with $v_{k}=g_{k}$ for every $k$. Now let $\varepsilon>0$ and $J$ be as above, and $j\ge J$; by the second part of $(*)$, $(**)$ and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion, $d(v^{j},v)^{2}=\sum_{k=1}^{\infty}\lVert v^{j}_{k}-g_{k}\rVert_{L^{2}(\mu)}^{2}=\sup_{K}\sum_{k=1}^{K}\lVert v^{j}_{k}-g_{k}\rVert_{L^{2}(\mu)}^{2}\le\varepsilon^{2}$, so $d(v^{j},v)\le\varepsilon$ for every $j\ge J$. Applying this with $\varepsilon/2$ in place of $\varepsilon$, every $\varepsilon>0$ admits $J$ with $d(v^{j},v)\le\varepsilon/2<\varepsilon$ for all $j\ge J$. Hence $(v^{j})$ converges to $v$, the metric space $(L^{2}(\mu;E),d)$ is complete, and $L^{2}(\mu;E)$ is a real Hilbert space by Real Hilbert Space §hilbert.