Proof of The Wiener Chaos Decomposition: Cylindrical Hermite Polynomials Are Dense and the Gaussian Lebesgue Space Is the Orthogonal Sum of the Chaoses

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

Write $L^{2}=L^{2}(\gamma_{c})$, a real Hilbert space with inner product $\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{L^{2}(\gamma_{c})}$ whose norm is $\lVert\cdot\rVert_{2}$, by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue and The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product; its distance is $d(F,G)=\lVert F-G\rVert_{2}$, and closures, closedness and convergence refer to the metric space $(L^{2},d)$ as in Real Hilbert Space §topology. Each $H_{\alpha}$ lies in $L^{2}$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical. For $m\in\mathbb{N}_{0}$ let $S_{m}$ be the linear span (in the sense of The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space) of $\{H_{\alpha}:\alpha\in\mathcal{A}_{m}\}$ and $S_{\le m}$ that of $\{H_{\alpha}:|\alpha|\le m\}$, so that $\mathcal{H}_{m}$ and $\mathcal{H}_{\le m}$ are their closures by The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §chaos and The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §chaos-up-to. For a closed linear subspace $N$, $P_{N}$ is the orthogonal projection of Real Hilbert Spaces: Standing Notation and Background §projections, the nearest-point map of Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space §existence; in particular $P_{N}F\in N$ and $\lVert F-P_{N}F\rVert_{2}\le\lVert F-G\rVert_{2}$ for all $G\in N$.

Step 1 (Spans and their closures are linear subspaces). Let $B\subseteq\mathcal{A}$ and let $S$ be the linear span of $\{H_{\alpha}:\alpha\in B\}$. It contains the zero vector $0$ of $L^{2}$. Let $t\in\mathbb{R}$. By claim 4 of Elementary Identities in a Vector Space, $t\,0=0$; and for an element $\sum_{i=1}^{r}t_{i}F_{i}$ of $S$, claim 3 of Properties of Finite Sums of Vectors and the axiom $t(t_{i}F_{i})=(tt_{i})F_{i}$ of Vector Space over a Field give $t\sum_{i=1}^{r}t_{i}F_{i}=\sum_{i=1}^{r}(tt_{i})F_{i}\in S$. For sums, $0+G=G+0=G$ for every $G\in S$; so let $u=\sum_{i=1}^{r}u_{i}$ and $u'=\sum_{j=1}^{r'}u'_{j}$ with $u_{i}=t_{i}F_{i}$, $u'_{j}=t'_{j}F'_{j}$, all $F_{i},F'_{j}$ in $\{H_{\alpha}:\alpha\in B\}$, and let $w\in(L^{2})^{r+r'}$ be the concatenation, $w_{i}=u_{i}$ for $i\le r$ and $w_{r+j}=u'_{j}$ for $j\le r'$. Let $A$ be the set of $l\in\mathbb{N}$ such that, if $l\le r'$, then $\sum_{i=1}^{r+l}w_{i}=u+\sum_{j=1}^{l}u'_{j}$. By claim 1 of Properties of Finite Sums of Vectors (restriction: the sums of $w$ up to $r$ are those of $(u_{i})_{i\le r}$; recursion: $\sum_{i=1}^{r+l+1}w_{i}=\sum_{i=1}^{r+l}w_{i}+w_{r+l+1}$), $\sum_{i=1}^{r+1}w_{i}=u+u'_{1}=u+\sum_{j=1}^{1}u'_{j}$, so $1\in A$; and if $l\in A$ and $l+1\le r'$, then $\sum_{i=1}^{r+l+1}w_{i}=\bigl(u+\sum_{j=1}^{l}u'_{j}\bigr)+u'_{l+1}=u+\sum_{j=1}^{l+1}u'_{j}$ by associativity (Vector Space over a Field) and claim 1 again, so $l+1\in A$ (trivially if $r'<l+1$). Hence $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers, and $l=r'$ gives $u+u'=\sum_{i=1}^{r+r'}w_{i}\in S$. So $S$ is a linear subspace. Let $\bar{S}$ be its closure. It is closed and contains $S$, by claims 2 and 1 of The Closure is the Smallest Closed Superset, so it contains the zero vector. Let $F,G\in\bar{S}$ and $t\in\mathbb{R}$. By Sequential Characterization of the Closure in a Metric Space there are sequences $(F_{j})$, $(G_{j})$ in $S$ converging to $F$ and $G$; then $F_{j}+G_{j}\in S$ and $tF_{j}\in S$ converge to $F+G$ and $tF$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits, so $F+G,tF\in\bar{S}$ by Sequential Characterization of the Closure in a Metric Space again. Thus $\bar{S}$ is a closed linear subspace (Real Hilbert Space §closed-subspace). In particular $\mathcal{H}_{m}$ and $\mathcal{H}_{\le m}$ are closed linear subspaces of $L^{2}$ for every $m\in\mathbb{N}_{0}$, and the projections $J_{m}=P_{\mathcal{H}_{m}}$ are defined.

Step 2 (Orthogonality to a closed span). Let $B\subseteq\mathcal{A}$, let $\bar{S}$ be as in Step 1, and let $F\in L^{2}$ with $\langle F,H_{\alpha}\rangle=0$ for every $\alpha\in B$. Then $\langle F,G\rangle=0$ for every $G\in\bar{S}$. Indeed, for $G=\sum_{i=1}^{r}t_{i}H_{\alpha_{i}}\in S$ we get $\langle F,G\rangle=\sum_{i=1}^{r}t_{i}\langle F,H_{\alpha_{i}}\rangle=0$ by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and claim 7 of Properties of Finite Sums, every term being $t_{i}\cdot0=0$, and $\langle F,0\rangle=0$ by Elementary Identities in a Real Inner Product Space §zero; for $G\in\bar{S}$ take $G_{j}\in S$ converging to $G$ (Sequential Characterization of the Closure in a Metric Space), so that $\langle F,G\rangle=\lim_{j}\langle F,G_{j}\rangle=0$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, applied to the constant sequence with every term $F$ and to $(G_{j})$.

Step 3 (The chaos of order $0$). Let $0_{\mathcal{A}}$ be the zero multi-index, the sequence all of whose terms are $0$; $1$ is a length bound for it (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §multi-indices), and its order is $\sum_{k=1}^{1}0=0$ by Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order and claim 1 of Properties of Finite Sums, so $0_{\mathcal{A}}\in\mathcal{A}_{0}$. Conversely, let $\alpha\in\mathcal{A}_{0}$ have length bound $n$. Then $\sum_{k=1}^{n}\alpha_{k}=|\alpha|=0$ with nonnegative summands, so $\alpha_{k}=0$ for every $k\le n$ by claim 5 of Properties of Finite Sums, and $\alpha_{k}=0$ for $k>n$ as $n$ is a length bound; thus $\alpha=0_{\mathcal{A}}$. So $\mathcal{A}_{0}=\{0_{\mathcal{A}}\}$, and with the length bound $1$, $H_{0_{\mathcal{A}}}(x)=H^{c_{1}}_{0}(x_{1})=1$ by The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space §hermite and Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §low-orders. Write $\mathbf{1}$ for the class of the constant function $1$. For $r\in\mathbb{N}$ and $t_{1},\dots,t_{r}\in\mathbb{R}$ we have $\sum_{i=1}^{r}t_{i}\mathbf{1}=\bigl(\sum_{i=1}^{r}t_{i}\bigr)\mathbf{1}$: the set of $l\in\mathbb{N}$ such that, if $l\le r$, then $\sum_{i=1}^{l}t_{i}\mathbf{1}=\bigl(\sum_{i=1}^{l}t_{i}\bigr)\mathbf{1}$ contains $1$ by claim 1 of Properties of Finite Sums of Vectors and claim 1 of Properties of Finite Sums, and contains $l+1$ whenever it contains $l$, by the recursions in those claims and the axiom $(a+b)v=av+bv$ of Vector Space over a Field; so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers. Hence $S_{0}=\{t\mathbf{1}:t\in\mathbb{R}\}$, the zero vector being $0\,\mathbf{1}$ (claim 3 of Elementary Identities in a Vector Space); the class of the constant function $t$ is $t\mathbf{1}$ by The Lebesgue Space of Power-Integrable Functions §space, so $S_{0}$ is the set of classes of constant functions. Since $\gamma_{c}\in\mathcal{P}(X)$ by Diagonal Gaussian Measures on a Hilbert Space §measure, $\gamma_{c}(X)=1$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, so The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space give $\lVert\mathbf{1}\rVert_{2}^{2}=\langle\mathbf{1},\mathbf{1}\rangle=\int_{X}1\cdot1\,d\gamma_{c}=\gamma_{c}(X)=1$, so the $1$-tuple $(\mathbf{1})$ is orthonormal (Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal) and $S_{0}$ is its span, which is closed by Projection onto the Span of an Orthonormal Tuple, and Coordinates on a Finite-Dimensional Subspace §closed. By claim 4 of The Closure is the Smallest Closed Superset, $\mathcal{H}_{0}=S_{0}$ is the set of classes of constant functions.

Step 4 (Orthogonality of different chaoses). Let $m\ne m'$ in $\mathbb{N}_{0}$. For $\alpha\in\mathcal{A}_{m}$ and $\beta\in\mathcal{A}_{m'}$ we have $|\alpha|\ne|\beta|$, so $\alpha\ne\beta$ and $\langle H_{\alpha},H_{\beta}\rangle=0$ by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §orthogonality. Fix $\alpha\in\mathcal{A}_{m}$; Step 2 (with $B=\mathcal{A}_{m'}$, $F=H_{\alpha}$, and the symmetry of the inner product) gives $\langle G,H_{\alpha}\rangle=0$ for every $G\in\mathcal{H}_{m'}$. Now fix $G\in\mathcal{H}_{m'}$; Step 2 (with $B=\mathcal{A}_{m}$, $F=G$) gives $\langle G,F'\rangle=0$ for every $F'\in\mathcal{H}_{m}$. So every element of $\mathcal{H}_{m}$ is orthogonal to every element of $\mathcal{H}_{m'}$.

Step 5 (The projection onto $\mathcal{H}_{\le m}$). Let $F\in L^{2}$, $m\in\mathbb{N}_{0}$ and $S=\sum_{j=0}^{m}J_{j}F$. For $j\le m$, every $H_{\alpha}$ with $\alpha\in\mathcal{A}_{j}$ lies in $S_{\le m}$, a linear subspace (Step 1), so $S_{j}\subseteq S_{\le m}$ and $\mathcal{H}_{j}\subseteq\mathcal{H}_{\le m}$ by claim 5 of The Closure is the Smallest Closed Superset. To handle the sum from $0$, let $b\in(L^{2})^{m+1}$ have components $b_{k}=J_{k-1}F$ ($k\in[m+1]$). By Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers, $S=J_{0}F$ if $m=0$ and $S=J_{0}F+\sum_{i=1}^{m}J_{i}F$ if $m\ge1$. In the first case $S=b_{1}=\sum_{k=1}^{1}b_{k}$ by claim 1 of Properties of Finite Sums of Vectors; in the second, Extraction of a Summand from a Finite Sum of Vectors (with $n=m$ and the index $1$) gives $\sum_{k=1}^{m+1}b_{k}=\sum_{k=1}^{m}b_{k+1}+b_{1}=\sum_{i=1}^{m}J_{i}F+J_{0}F$; so in both cases $S=\sum_{k=1}^{m+1}b_{k}$, by commutativity of addition. Each $b_{k}=J_{k-1}F=P_{\mathcal{H}_{k-1}}F$ lies in $\mathcal{H}_{k-1}$, which is contained in $\mathcal{H}_{\le m}$ by the preceding inclusion with $j=k-1\le m$; and $\mathcal{H}_{\le m}$ is a linear subspace of $L^{2}$ by Step 1. So claim 3 of The Span of a Finite Family is the Smallest Subspace Containing It, applied with $W=\mathcal{H}_{\le m}$ to the $(m+1)$-tuple $b$, gives $\operatorname{span}(b)\subseteq\mathcal{H}_{\le m}$. Since $\sum_{k=1}^{m+1}b_{k}=\sum_{k=1}^{m+1}1\,b_{k}$ by the axiom $1v=v$ of Vector Space over a Field, $S$ belongs to the span of $b$ (take all coefficients equal to $1$), hence $S\in\mathcal{H}_{\le m}$. Let $\alpha\in\mathcal{A}$ with $j=|\alpha|\le m$. If $m=0$, then $j=0$ and $\langle F-S,H_{\alpha}\rangle=\langle F-J_{j}F,H_{\alpha}\rangle$. If $m\ge1$, Extraction of a Summand from a Finite Sum of Vectors (with $n=m$ and the index $j+1$) gives $S=\sum_{k=1}^{m}b^{(j+1)}_{k}+J_{j}F$, where each component $b^{(j+1)}_{k}$ is $J_{i}F$ for some $i\in\{0,\dots,m\}$ with $i\ne j$; so, by bilinearity (Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §sums, Elementary Identities in a Real Inner Product Space §bilinear),

$$\langle F-S,H_{\alpha}\rangle=\langle F-J_{j}F,H_{\alpha}\rangle-\sum_{k=1}^{m}\bigl\langle b^{(j+1)}_{k},H_{\alpha}\bigr\rangle .$$

The first term vanishes because $F-J_{j}F\in\mathcal{H}_{j}^{\perp}$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation and $H_{\alpha}\in S_{j}\subseteq\mathcal{H}_{j}$; each term of the sum vanishes by Step 4, since $b^{(j+1)}_{k}=J_{i}F\in\mathcal{H}_{i}$ and $H_{\alpha}\in\mathcal{H}_{j}$ with $i\ne j$, so the sum is $0$ by claim 7 of Properties of Finite Sums. By Step 2 with $B=\{\alpha:|\alpha|\le m\}$, $F-S\in\mathcal{H}_{\le m}^{\perp}$, so $S=P_{\mathcal{H}_{\le m}}F$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation. Steps 1 and 3 to 5 prove claim 1.

Step 6 (Claim 2). Let $F\in L^{2}$ and $\varepsilon>0$. Since $\gamma_{c}\in\mathcal{P}(X)$, Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density and Sequential Characterization of the Closure in a Metric Space give $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert F-\varphi\rVert_{2}<\varepsilon/2$: $F$ lies in the closure of the set of classes of such functions, so some sequence of such classes converges to $F$, and one of its terms, the class of some $\varphi$, has distance less than $\varepsilon/2$ from $F$; here $\varphi$ is $2$-integrable and its class lies in $L^{2}$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable. By Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, $\varphi=\psi\circ p_{n}$ with $n\in\mathbb{N}$ and $\psi\in C^{1}_{b}(\mathbb{R}^{n})$. The function $\psi$ is continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, hence Borel by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space, and bounded by some $b\ge0$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded). So $\psi^{2}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with $0\le\psi^{2}\le b^{2}$. Since $(p_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}$ (Diagonal Gaussian Measures on a Hilbert Space §measure) is the image measure of $\gamma_{c}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward), claim 1 of Image Measures, Measures with Densities, and Change of Variables gives $\gamma_{c^{(n)}}(\mathbb{R}^{n})=\gamma_{c}(X)=1$, and claims 6(b) and 6(c) of Borel Measurability and Bounded Integration on a Metric Space give $\int_{\mathbb{R}^{n}}\psi^{2}\,d\gamma_{c^{(n)}}\le b^{2}<\infty$. The truncation $c^{(n)}$ is a variance vector (Variance Sequences and Their Truncations §truncations), so Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space §dense gives a polynomial function $P$ on $\mathbb{R}^{n}$ with $\int_{\mathbb{R}^{n}}(\psi-P)^{2}\,d\gamma_{c^{(n)}}<\varepsilon^{2}/4$; $P$ is Borel with $\int P^{2}\,d\gamma_{c^{(n)}}<\infty$ by Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space §integrable. Since $p_{n}$ is Borel (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity), $P\circ p_{n}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. The functions $P^{2}$ and $(\psi-P)^{2}$ are nonnegative and Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, so claim 2 of Image Measures, Measures with Densities, and Change of Variables, applied to them with $(p_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}$, gives $\int_{X}(P\circ p_{n})^{2}\,d\gamma_{c}=\int_{\mathbb{R}^{n}}P^{2}\,d\gamma_{c^{(n)}}<\infty$ and

$$\lVert\varphi-P\circ p_{n}\rVert_{2}^{2}=\int_{X}(\psi\circ p_{n}-P\circ p_{n})^{2}\,d\gamma_{c}=\int_{\mathbb{R}^{n}}(\psi-P)^{2}\,d\gamma_{c^{(n)}}<\frac{\varepsilon^{2}}{4},$$

using Power-Integrable Functions and the p-Seminorm §seminorm and The Lebesgue Space of Power-Integrable Functions §norm; so $\lVert\varphi-P\circ p_{n}\rVert_{2}<\varepsilon/2$. Write $P=\sum_{i=1}^{r}t_{i}\mu_{i}$ with monomials $\mu_{i}(y)=\prod_{k=1}^{n}y_{k}^{\iota_{i,k}}$ of exponents $\iota_{i}$ of length $n$ (Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space). Let $\alpha^{i}\in\mathcal{A}$ have terms $\iota_{i,1},\dots,\iota_{i,n}$ followed by zeros, so that $n$ is a length bound for $\alpha^{i}$ and $\mu_{i}(p_{n}(x))=\prod_{k=1}^{n}x_{k}^{\alpha^{i}_{k}}$. By Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §monomials each $\mu_{i}\circ p_{n}$ is a finite linear combination of functions $H_{\beta}$, hence so is $P\circ p_{n}=\sum_{i}t_{i}\,\mu_{i}\circ p_{n}$, and by the operations of The Lebesgue Space of Power-Integrable Functions §space its class $G$ is the same linear combination of the classes $H_{\beta}$; thus $G\in\mathcal{H}_{\mathrm{fin}}$ (The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §span). By the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle, $\lVert F-G\rVert_{2}\le\lVert F-\varphi\rVert_{2}+\lVert\varphi-G\rVert_{2}<\varepsilon$. This proves claim 2.

Step 7 (Claim 3). Let $F\in L^{2}$ and, for $M\in\mathbb{N}_{0}$, $S_{M}^{F}=\sum_{m=0}^{M}J_{m}F$ (a sum from $0$ as in Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers), which equals $P_{\mathcal{H}_{\le M}}F$ by Step 5. Let $\varepsilon>0$ and take $G\in\mathcal{H}_{\mathrm{fin}}$ with $\lVert F-G\rVert_{2}<\varepsilon$ (Step 6). Either $G$ is the zero vector, and we put $M_{0}=1$, or $G=\sum_{i=1}^{r}t_{i}H_{\alpha_{i}}$, and we let $M_{0}$ be a largest component of the $(r+1)$-tuple $(1,|\alpha_{1}|,\dots,|\alpha_{r}|)$ of real numbers, which exists by Greatest Element of a Finite Family in a Totally Ordered Set; in either case $M_{0}\in\mathbb{N}$. For $M\in\mathbb{N}$ with $M\ge M_{0}$ we have $|\alpha_{i}|\le M$ for every $i$, so $G$ is an element of the span $S_{\le M}$ as defined, and $G\in S_{\le M}\subseteq\mathcal{H}_{\le M}$ (claim 1 of The Closure is the Smallest Closed Superset), so by the nearest-point property $\lVert F-S^{F}_{M}\rVert_{2}\le\lVert F-G\rVert_{2}<\varepsilon$. Hence $(\lVert F-S^{F}_{M}\rVert_{2})_{M\in\mathbb{N}}$, a sequence of nonnegative reals, converges to $0$.

By Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras with the closed linear subspace $\mathcal{H}_{\le M}$, $\lVert F\rVert_{2}^{2}=\lVert S^{F}_{M}\rVert_{2}^{2}+\lVert F-S^{F}_{M}\rVert_{2}^{2}$. Put $x_{m}=\lVert J_{m}F\rVert_{2}^{2}$ for $m\in\mathbb{N}_{0}$. For $M\in\mathbb{N}$ we have

$$S^{F}_{M}=S^{F}_{M-1}+J_{M}F,\qquad\sum_{m=0}^{M}x_{m}=\sum_{m=0}^{M-1}x_{m}+x_{M}:$$

by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers, $S^{F}_{M}=J_{0}F+\sum_{m=1}^{M}J_{m}F$; if $M=1$, $\sum_{m=1}^{1}J_{m}F=J_{1}F$ and $S^{F}_{0}=J_{0}F$, while if $M\ge2$, the recursion of claim 1 of Properties of Finite Sums of Vectors and associativity give $J_{0}F+\sum_{m=1}^{M}J_{m}F=\bigl(J_{0}F+\sum_{m=1}^{M-1}J_{m}F\bigr)+J_{M}F=S^{F}_{M-1}+J_{M}F$; the same argument with claim 1 of Properties of Finite Sums gives the identity for the $x_{m}$. Let $A$ be the set of $M\in\mathbb{N}$ such that $\lVert S^{F}_{M-1}\rVert_{2}^{2}=\sum_{m=0}^{M-1}x_{m}$. Then $1\in A$, as both sides equal $x_{0}$ by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers. Let $M\in A$. By Step 5 (with $M-1$ in place of $m$), $S^{F}_{M-1}=\sum_{k=1}^{M}b_{k}$ with $b_{k}=J_{k-1}F$, so $\langle S^{F}_{M-1},J_{M}F\rangle=\sum_{k=1}^{M}\langle J_{k-1}F,J_{M}F\rangle=0$ by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §sums, Step 4 (as $k-1\ne M$) and claim 7 of Properties of Finite Sums. Hence Elementary Identities in a Real Inner Product Space §expansion and the displayed identities give $\lVert S^{F}_{M}\rVert_{2}^{2}=\lVert S^{F}_{M-1}\rVert_{2}^{2}+2\langle S^{F}_{M-1},J_{M}F\rangle+\lVert J_{M}F\rVert_{2}^{2}=\sum_{m=0}^{M-1}x_{m}+x_{M}=\sum_{m=0}^{M}x_{m}$, that is, $M+1\in A$. So $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers, and $\lVert S^{F}_{M}\rVert_{2}^{2}=\sum_{m=0}^{M}x_{m}$ for every $M\in\mathbb{N}_{0}$, since $M+1\in A$. Then $\sum_{m=0}^{M}x_{m}=\lVert F\rVert_{2}^{2}-\lVert F-S^{F}_{M}\rVert_{2}^{2}$, which converges to $\lVert F\rVert_{2}^{2}$ as $M\to\infty$ by Arithmetic of Limits of Real Sequences §products and Arithmetic of Limits of Real Sequences §scalar. By Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers, $\sum_{m=0}^{M}x_{m}=x_{0}+\sum_{m=1}^{M}x_{m}$, so the partial sums of $\sum_{m=1}^{\infty}x_{m}$ converge to $\lVert F\rVert_{2}^{2}-x_{0}$ (Arithmetic of Limits of Real Sequences §scalar); that series converges in the sense of Series of Real Numbers §convergent with this sum, and by Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers the series $\sum_{m=0}^{\infty}x_{m}$ converges with sum $x_{0}+(\lVert F\rVert_{2}^{2}-x_{0})=\lVert F\rVert_{2}^{2}$. This proves claim 3.