Proof of The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It

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

Throughout, $d\in\mathbb{N}$ and the notation of the statement are fixed; natural numbers are read in $\mathbb{R}$ through the canonical map, as in The Real Numbers: Standing Notation and Background §numbers. We write $\theta=\theta_{d}$ and $r=\tfrac{1}{3072}$, so that $0\le r<1$. The clauses are proved in the order 3, 1, 2, 4, 5, 6, 7, 8 (Steps 2 to 9), after preliminaries (Steps 0 and 1).

Step 0 (Preliminaries).

(0a) Complex arithmetic. Let $z,\zeta\in\mathbb{C}$ and $s\in\mathbb{R}$. By Real and Imaginary Parts of a Complex Number, $z=\operatorname{Re}z+(\operatorname{Im}z)i$, and by claim 3 of Canonical Form and Arithmetic of Complex Numbers a complex number is determined by its real and imaginary parts. Writing $s=s+0i$ and using claims 3 and 4 of Canonical Form and Arithmetic of Complex Numbers, one obtains $\operatorname{Re}(z+\zeta)=\operatorname{Re}z+\operatorname{Re}\zeta$, $\operatorname{Im}(z+\zeta)=\operatorname{Im}z+\operatorname{Im}\zeta$, $\operatorname{Re}(sz)=s\operatorname{Re}z$, $\operatorname{Im}(sz)=s\operatorname{Im}z$, $\operatorname{Re}s=s$ and $\operatorname{Im}s=0$; in particular, with $z-\zeta=z+(-1)\zeta$, also $\operatorname{Re}(z-\zeta)=\operatorname{Re}z-\operatorname{Re}\zeta$ and $\operatorname{Im}(z-\zeta)=\operatorname{Im}z-\operatorname{Im}\zeta$. By Complex Conjugate and claim 3 of Canonical Form and Arithmetic of Complex Numbers, $\operatorname{Re}\overline{z}=\operatorname{Re}z$ and $\operatorname{Im}\overline{z}=-\operatorname{Im}z$. By Modulus of a Complex Number, $0\le|z|$ and $|z|^{2}=(\operatorname{Re}z)^{2}+(\operatorname{Im}z)^{2}$. By claim 8 of Properties of Complex Conjugation and Modulus the modulus of a real number is its absolute value, so claim 6 of that lemma gives $|\operatorname{Re}z|\le|z|$ and $|\operatorname{Im}z|\le|z|$. By claims 8 and 4 of that lemma, $|-1|=1$ and $|sz|=|s|\,|z|$, $|\zeta z|=|\zeta|\,|z|$; hence $|-z|=|(-1)z|=|z|$ and $|z-\zeta|=|\zeta-z|$. By claim 3 of that lemma $\overline{z}\,z=z\overline{z}=|z|^{2}$, a real number, and by claim 1 of that lemma $\overline{s}=s$, $\overline{z+\zeta}=\overline{z}+\overline{\zeta}$, $\overline{z\zeta}=\overline{z}\,\overline{\zeta}$ and $\overline{\overline{z}}=z$. By claim 2 of that lemma $z+\overline{z}=2\operatorname{Re}z$. Finally, by the triangle inequality (claim 7 of that lemma), $|z-\zeta|\le|z|+|\zeta|$.

(0b) Real sequences. By claim 1 of Convergence and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, a sequence $(a_{m})_{m\in\mathbb{N}}$ of real numbers converges to $A\in\mathbb{R}$ in the sense of Limit of a Sequence of Real Numbers, which is the sense used in Arithmetic of Limits of Real Sequences, Order Properties of Limits of Real Sequences, Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits and The Diagonal Subsequence Lemma for Bounded Real Arrays, if and only if it converges to $A$ in $(\mathbb{R},d_{\mathbb{R}})$, that is, for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|a_{m}-A|<\varepsilon$ for every $m\in\mathbb{N}$ with $N\le m$. We use both descriptions. Consequently a constant sequence converges to its value; limits are unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences; and, by A Subsequence of a Convergent Sequence Has the Same Limit in $(\mathbb{R},d_{\mathbb{R}})$, every subsequence of a sequence converging to $A$ converges to $A$. For $N,N'\in\mathbb{N}$ there is $M\in\mathbb{N}$ with $N\le M$ and $N'\le M$: by claims 3 and 1 of Properties of the Order on the Natural Numbers, take $M=N'$ if $N<N'$ or $N=N'$, and $M=N$ if $N'<N$.

(0c) Eventual bounds pass to limits. Let $(a_{m})_{m\in\mathbb{N}}$ be a sequence of real numbers converging to $A$, let $C\in\mathbb{R}$ and $N\in\mathbb{N}$, and suppose $a_{m}\le C$ for every $m\ge N$. Then $A\le C$. Otherwise $C<A$; put $\eta=A-C>0$, choose $N'$ with $|a_{m}-A|<\eta$ for all $m\ge N'$ by (0b), and $M$ with $N\le M$ and $N'\le M$ by (0b). By Absolute Value in an Ordered Field, $A-a_{M}\le|a_{M}-A|<A-C$, so $C<a_{M}$, contradicting $a_{M}\le C$.

(0d) Complex sequences. Convergence of a sequence of complex numbers means convergence in $(\mathbb{C},d_{\mathbb{C}})$, $d_{\mathbb{C}}(z,z')=|z-z'|$, a metric by claim 9 of Properties of Complex Conjugation and Modulus. Let $(z_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathbb{C}$ and $a\in\mathbb{C}$.

(i) $(z_{m})$ converges to $a$ if and only if $(\operatorname{Re}z_{m})$ converges to $\operatorname{Re}a$ and $(\operatorname{Im}z_{m})$ converges to $\operatorname{Im}a$. If $(z_{m})$ converges to $a$ and $\varepsilon>0$, choose $N$ with $|z_{m}-a|<\varepsilon$ for $m\ge N$; by (0a), $|\operatorname{Re}z_{m}-\operatorname{Re}a|=|\operatorname{Re}(z_{m}-a)|\le|z_{m}-a|<\varepsilon$, and likewise for $\operatorname{Im}$; so both real sequences converge by (0b). Conversely, suppose both real sequences converge as stated, and let $\varepsilon>0$. By (0a), $|z_{m}-a|^{2}=(\operatorname{Re}z_{m}-\operatorname{Re}a)^{2}+(\operatorname{Im}z_{m}-\operatorname{Im}a)^{2}$, which converges to $0\cdot0+0\cdot0=0$ by claims 3, 2 and 1 of Arithmetic of Limits of Real Sequences and (0b). Since $\varepsilon^{2}>0$, there is $N$ with $|z_{m}-a|^{2}=\bigl||z_{m}-a|^{2}-0\bigr|<\varepsilon^{2}$ for $m\ge N$ (the number $|z_{m}-a|^{2}$ being nonnegative), and then $|z_{m}-a|<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both numbers being nonnegative.

(ii) If $(z_{m})$ converges to $a$ and to $a'$, then $a=a'$: by (i) and the uniqueness of real limits (0b), $\operatorname{Re}a=\operatorname{Re}a'$ and $\operatorname{Im}a=\operatorname{Im}a'$, so $a=a'$ by (0a). A constant sequence converges to its value, since $d_{\mathbb{C}}(a,a)=|0|=0$ by claim 3 of Properties of Complex Conjugation and Modulus.

(iii) If $(z_{m})$ converges to $a$ and $b\in\mathbb{C}$, then the real sequence $(|z_{m}-b|^{2})_{m}$ converges to $|a-b|^{2}$: by (0a), $|z_{m}-b|^{2}=(\operatorname{Re}z_{m}-\operatorname{Re}b)^{2}+(\operatorname{Im}z_{m}-\operatorname{Im}b)^{2}$, which by (i), (0b) and claims 3, 2 and 1 of Arithmetic of Limits of Real Sequences converges to $(\operatorname{Re}a-\operatorname{Re}b)^{2}+(\operatorname{Im}a-\operatorname{Im}b)^{2}=|a-b|^{2}$.

(iv) If $(z_{m})$ converges to $a$ and $\zeta\in\mathbb{C}$, then $(\overline{z_{m}})$ converges to $\overline{a}$ and $(\zeta z_{m})$ converges to $\zeta a$. Indeed, by (0a), $|\overline{z_{m}}-\overline{a}|=|\overline{z_{m}-a}|=|z_{m}-a|$, using claims 1 and 3 of Properties of Complex Conjugation and Modulus; and $|\zeta z_{m}-\zeta a|=|\zeta|\,|z_{m}-a|\le(|\zeta|+1)|z_{m}-a|$, so for $\varepsilon>0$ it suffices to take $N$ with $|z_{m}-a|<\varepsilon/(|\zeta|+1)$ for $m\ge N$.

(0e) Finite sums. Let $G$ be a nonempty finite set with $n$ elements and $\varphi:[n]\to G$ a bijection, so that $\sum_{x\in G}f(x)=\sum_{j=1}^{n}f(\varphi(j))$ for every map $f$ on $G$ with real or complex values (Sum over a Finite Index Set).

(i) If $h:G\to\mathbb{R}$ satisfies $0\le h(x)$ for every $x\in G$, then $h(x)\le\sum_{y\in G}h(y)$ for every $x\in G$ (write $x=\varphi(j)$ and apply claim 6 of Properties of Finite Sums), and if $\sum_{y\in G}h(y)=0$ then $h(x)=0$ for every $x\in G$ (claim 5 of that lemma).

(ii) If $f_{m},f:G\to\mathbb{R}$ ($m\in\mathbb{N}$) and $(f_{m}(x))_{m}$ converges to $f(x)$ for every $x\in G$, then $\bigl(\sum_{x\in G}f_{m}(x)\bigr)_{m}$ converges to $\sum_{x\in G}f(x)$: apply Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §limit, with $N=j=n$, to $a_{k,m}=f_{m}(\varphi(k))$.

(iii) If $f_{m},f:G\to\mathbb{C}$ ($m\in\mathbb{N}$) and $(f_{m}(x))_{m}$ converges to $f(x)$ in $(\mathbb{C},d_{\mathbb{C}})$ for every $x\in G$, then $\bigl(\sum_{x\in G}f_{m}(x)\bigr)_{m}$ converges to $\sum_{x\in G}f(x)$ in $(\mathbb{C},d_{\mathbb{C}})$. Indeed, by claims 3 and 4 of Properties of a Sum over a Finite Index Set, $\sum_{x\in G}f_{m}(x)-\sum_{x\in G}f(x)=\sum_{x\in G}\bigl(f_{m}(x)-f(x)\bigr)$, whose modulus is at most $s_{m}=\sum_{x\in G}|f_{m}(x)-f(x)|$ by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus. For each $x$ the real sequence $(|f_{m}(x)-f(x)|)_{m}$ converges to $0$ by (0b), since $\bigl||f_{m}(x)-f(x)|-0\bigr|=d_{\mathbb{C}}(f_{m}(x),f(x))$; so $(s_{m})$ converges to $\sum_{x\in G}0=0$ by (ii) and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing. Given $\varepsilon>0$, choose $N$ with $s_{m}<\varepsilon$ for $m\ge N$; then $\bigl|\sum_{x\in G}f_{m}(x)-\sum_{x\in G}f(x)\bigr|\le s_{m}<\varepsilon$ for $m\ge N$.

Step 1 (The gauge space pointwise). For $k\in\mathbb{N}$ the set $W^{\circ}_{d,k}$ is nonempty and finite by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite, and every $w\in W^{\circ}_{d}$ is either $\varnothing$ or lies in $W^{\circ}_{d,k}$ for exactly one $k\in\mathbb{N}$, its length (Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §length); for such $w$ one has $|w|=k$ and $c_{w}=\theta^{k}$, while $c_{\varnothing}=1$ and $|\varnothing|=0$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights). For a map $z:W^{\circ}_{d}\to\mathbb{C}$ and $k,K\in\mathbb{N}$ put

$$Z_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}|z(w)|^{2},\qquad T_{K}(z)=c_{\varnothing}|z(\varnothing)|^{2}+\sum_{k=1}^{K}Z_{k}.$$

Every $c_{w}$ is positive (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights) and $0\le|z(w)|^{2}$, so every term $c_{w}|z(w)|^{2}$ is nonnegative and $0\le Z_{k}$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. For $z\in E_{d}$, by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm, $0\le\lVert z\rVert_{d}$ and

$$\lVert z\rVert_{d}^{2}=|z(\varnothing)|^{2}+\sum_{k=1}^{\infty}Z_{k}.$$

(1a) Vector space. By The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space, $E_{d}$ is closed under pointwise sums and pointwise real multiples. The zero map $0$ lies in $E_{d}$: all its terms $c_{w}|0|^{2}$ vanish, so its block sums vanish by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, the partial sums of their series vanish by Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §nonnegative, and the series converges to $0$ by (0b); thus $\lVert0\rVert_{d}^{2}=0$ and $\lVert0\rVert_{d}=0$. By condition 1 of The Complex Numbers and claim 1 of Canonical Form and Arithmetic of Complex Numbers, sums and products of real numbers, and the identities $0$ and $1$, are the same in $\mathbb{R}$ and in $\mathbb{C}$; hence conditions 1, 2, 5, 6, 7 and 8 of Vector Space over a Field over the field $\mathbb{R}$ hold for the pointwise operations, value by value, by the field axioms of $\mathbb{C}$; condition 3 holds with the zero map; and condition 4 holds with $(-1)x$, since $x(w)+(-1)x(w)=0$ for every $w$. So $E_{d}$ is a vector space over $\mathbb{R}$ whose zero vector is the zero map (unique by claim 1 of Elementary Identities in a Vector Space), and by claims 2 and 5 of that lemma the additive inverse of $y\in E_{d}$ is $-y=(-1)y$. Hence the vector-space difference $x-y=x+(-y)$ is the difference $x+(-1)y$ of The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space, and $(x-y)(w)=x(w)-y(w)$ for every $w\in W^{\circ}_{d}$.

(1b) Coordinates are dominated by the norm. Let $z\in E_{d}$, $K\in\mathbb{N}$ and $w\in W^{\circ}_{d}$. Then $T_{K}(z)\le\lVert z\rVert_{d}^{2}$ and $c_{w}|z(w)|^{2}\le\lVert z\rVert_{d}^{2}$. Indeed, by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates (the terms $Z_{k}$ being nonnegative and their series convergent), $0\le\sum_{k=1}^{K}Z_{k}\le\sum_{k=1}^{\infty}Z_{k}$, which gives the first inequality and also $0\le\sum_{k=1}^{\infty}Z_{k}$, whence $c_{\varnothing}|z(\varnothing)|^{2}\le\lVert z\rVert_{d}^{2}$. If $w\in W^{\circ}_{d,k}$, then $c_{w}|z(w)|^{2}\le Z_{k}$ by (0e)(i), $Z_{k}\le\sum_{j=1}^{k}Z_{j}$ by claim 6 of Properties of Finite Sums, $\sum_{j=1}^{k}Z_{j}\le T_{k}(z)$ because $0\le c_{\varnothing}|z(\varnothing)|^{2}$, and $T_{k}(z)\le\lVert z\rVert_{d}^{2}$.

(1c) Coordinate control. Let $w\in W^{\circ}_{d}$ and let $\varepsilon>0$ be real. The number $c_{w}\varepsilon^{2}$ is positive; let $\eta_{w}(\varepsilon)$ be its nonnegative square root (Existence and Uniqueness of the Nonnegative Square Root), which is nonzero because its square is nonzero, hence positive. If $z\in E_{d}$ and $\lVert z\rVert_{d}<\eta_{w}(\varepsilon)$, then $|z(w)|<\varepsilon$: by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $\lVert z\rVert_{d}^{2}<\eta_{w}(\varepsilon)^{2}=c_{w}\varepsilon^{2}$; by (1b), $c_{w}|z(w)|^{2}<c_{w}\varepsilon^{2}$; multiplying by the positive number $c_{w}^{-1}$ gives $|z(w)|^{2}<\varepsilon^{2}$, and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|z(w)|<\varepsilon$.

Step 2 (Clause 3: expansion of the inner product).

(2a) A pointwise identity. For $a,b\in\mathbb{C}$,

$$|a+b|^{2}-|a-b|^{2}=4\operatorname{Re}\bigl(\overline{a}\,b\bigr).$$

Indeed, by (0a), $|a+b|^{2}=(a+b)\overline{(a+b)}=(a+b)(\overline{a}+\overline{b})=a\overline{a}+a\overline{b}+b\overline{a}+b\overline{b}$ and, using $\overline{-b}=\overline{(-1)b}=(-1)\overline{b}$, $|a-b|^{2}=(a-b)(\overline{a}-\overline{b})=a\overline{a}-a\overline{b}-b\overline{a}+b\overline{b}$. Subtracting, and using commutativity in $\mathbb{C}$, $|a+b|^{2}-|a-b|^{2}=2\bigl(\overline{a}b+a\overline{b}\bigr)$. By (0a), $a\overline{b}=\overline{\overline{a}}\,\overline{b}=\overline{\overline{a}b}$, so $\overline{a}b+a\overline{b}=\overline{a}b+\overline{\overline{a}b}=2\operatorname{Re}(\overline{a}b)$, and the identity follows. Both sides are real.

(2b) The expansion. Let $x,y\in E_{d}$; then $x+y$ and $x-y$ lie in $E_{d}$ by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space. Let $S_{k}$ and $D_{k}$ be the $k$-th block sums of $w\mapsto c_{w}|x(w)+y(w)|^{2}$ and of $w\mapsto c_{w}|x(w)-y(w)|^{2}$, whose series converge. By claims 3 and 4 of Properties of a Sum over a Finite Index Set and (2a),

$$S_{k}-D_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}\bigl(|x(w)+y(w)|^{2}-|x(w)-y(w)|^{2}\bigr)=\sum_{w\in W^{\circ}_{d,k}}4c_{w}\operatorname{Re}\bigl(\overline{x(w)}\,y(w)\bigr)=4P_{k},$$

so $P_{k}=\tfrac14S_{k}+\bigl(-\tfrac14\bigr)D_{k}$. By Elementary Properties of Series of Real Numbers §linearity, $\sum_{k=1}^{\infty}P_{k}$ converges and equals $\tfrac14\bigl(\sum_{k=1}^{\infty}S_{k}-\sum_{k=1}^{\infty}D_{k}\bigr)$. By Step 1 (with $c_{\varnothing}=1$), The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §inner-product and (2a),

$$\langle x,y\rangle_{d}=\tfrac14\Bigl(|x(\varnothing)+y(\varnothing)|^{2}-|x(\varnothing)-y(\varnothing)|^{2}+\sum_{k=1}^{\infty}S_{k}-\sum_{k=1}^{\infty}D_{k}\Bigr)=\operatorname{Re}\bigl(\overline{x(\varnothing)}\,y(\varnothing)\bigr)+\sum_{k=1}^{\infty}P_{k}.$$

This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §expansion.

Step 3 (Clause 1: a real inner product space). By (1a), $E_{d}$ with the stated operations is a vector space over $\mathbb{R}$ with the zero map as zero vector, and $\langle x,y\rangle_{d}$ is a real number for all $x,y\in E_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §inner-product). For $x,y\in E_{d}$ write $P_{k}(x,y)$ for the number $P_{k}$ of Step 2, so that by Step 2 $\langle x,y\rangle_{d}=\operatorname{Re}\bigl(\overline{x(\varnothing)}\,y(\varnothing)\bigr)+\sum_{k=1}^{\infty}P_{k}(x,y)$. We verify conditions (a) to (d) of Real Inner Product Space §inner-product. Let $x,x',y\in E_{d}$ and $s\in\mathbb{R}$, and let $a,a',b\in\mathbb{C}$.

(a) By (0a), $\overline{\overline{a}b}=a\overline{b}=\overline{b}a$ and $\operatorname{Re}\overline{\zeta}=\operatorname{Re}\zeta$, so $\operatorname{Re}(\overline{a}b)=\operatorname{Re}(\overline{b}a)$. Applied at every $w$, this gives $P_{k}(x,y)=P_{k}(y,x)$ for every $k$ and equality of the terms at $\varnothing$, so $\langle x,y\rangle_{d}=\langle y,x\rangle_{d}$ by Step 2.

(b) By (0a), $\overline{a+a'}\,b=\overline{a}b+\overline{a'}b$ and $\operatorname{Re}$ is additive, so $\operatorname{Re}(\overline{a+a'}\,b)=\operatorname{Re}(\overline{a}b)+\operatorname{Re}(\overline{a'}b)$. Multiplying by $c_{w}$ and summing over $W^{\circ}_{d,k}$ (claim 3 of Properties of a Sum over a Finite Index Set) gives $P_{k}(x+x',y)=P_{k}(x,y)+P_{k}(x',y)$; by Elementary Properties of Series of Real Numbers §linearity and Step 2 (applied to $(x+x',y)$, $(x,y)$ and $(x',y)$), $\langle x+x',y\rangle_{d}=\langle x,y\rangle_{d}+\langle x',y\rangle_{d}$.

(c) By (0a), $\overline{sa}\,b=\overline{s}\,\overline{a}\,b=s\,\overline{a}b$ and $\operatorname{Re}(s\zeta)=s\operatorname{Re}\zeta$, so $\operatorname{Re}(\overline{sa}\,b)=s\operatorname{Re}(\overline{a}b)$. By claim 4 of Properties of a Sum over a Finite Index Set, $P_{k}(sx,y)=sP_{k}(x,y)$, and by Elementary Properties of Series of Real Numbers §linearity and Step 2, $\langle sx,y\rangle_{d}=s\langle x,y\rangle_{d}$.

(d) By (0a), $\overline{a}a=|a|^{2}$ is real, so $\operatorname{Re}(\overline{a}a)=|a|^{2}$. Hence $P_{k}(x,x)=X_{k}$, the $k$-th block sum of $w\mapsto c_{w}|x(w)|^{2}$, and Step 2 and Step 1 give

$$\langle x,x\rangle_{d}=|x(\varnothing)|^{2}+\sum_{k=1}^{\infty}X_{k}=\lVert x\rVert_{d}^{2}\ge0 .$$

Suppose $\langle x,x\rangle_{d}=0$. The two summands $|x(\varnothing)|^{2}$ and $\sum_{k=1}^{\infty}X_{k}$ are nonnegative (by (1b)) with sum $0$, so both vanish. For $k\in\mathbb{N}$, Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates and claim 6 of Properties of Finite Sums give $0\le X_{k}\le\sum_{j=1}^{k}X_{j}\le\sum_{j=1}^{\infty}X_{j}=0$, so $X_{k}=0$, and then $c_{w}|x(w)|^{2}=0$ for every $w\in W^{\circ}_{d,k}$ by (0e)(i). Since every $c_{w}$ is nonzero, $|x(w)|^{2}=0$ for every $w\in W^{\circ}_{d}$ (for $w=\varnothing$ as shown above), hence $|x(w)|=0$, since a field has no zero divisors, and $x(w)=0$ by claim 3 of Properties of Complex Conjugation and Modulus. Thus $x$ is the zero map, the zero vector of $E_{d}$.

So $\langle\cdot,\cdot\rangle_{d}$ is an inner product on $E_{d}$. Its norm at $x$ is the unique nonnegative real $\rho$ with $\rho^{2}=\langle x,x\rangle_{d}=\lVert x\rVert_{d}^{2}$; since $\lVert x\rVert_{d}$ is nonnegative, $\rho=\lVert x\rVert_{d}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. By (1a) and Real Inner Product Space §distance, the distance of $E_{d}$ is $d_{E}(x,y)=\lVert x-y\rVert_{d}$, with $(x-y)(w)=x(w)-y(w)$. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product.

Step 4 (Clause 2: completeness). By Real Hilbert Space §hilbert and Complete Metric Space we must show that every Cauchy sequence $(x_{n})_{n\in\mathbb{N}}$ in $(E_{d},d_{E})$ converges in $(E_{d},d_{E})$ to a point of $E_{d}$; $d_{E}$ is a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric.

(4a) The pointwise limit. Let $w\in W^{\circ}_{d}$ and $\varepsilon>0$. Choose $N$ with $d_{E}(x_{n},x_{m})=\lVert x_{n}-x_{m}\rVert_{d}<\eta_{w}(\varepsilon)$ for all $n,m\ge N$; by (1c) and (1a), $|x_{n}(w)-x_{m}(w)|=|(x_{n}-x_{m})(w)|<\varepsilon$ for $n,m\ge N$. So $(x_{n}(w))_{n}$ is a Cauchy sequence in $(\mathbb{C},d_{\mathbb{C}})$ and converges by The Complex Numbers are Complete in the Modulus Metric; let $x(w)$ be its limit, unique by (0d)(ii). This defines $x:W^{\circ}_{d}\to\mathbb{C}$.

(4b) The estimate. Let $\varepsilon>0$; choose $N$ with $\lVert x_{n}-x_{m}\rVert_{d}<\varepsilon/2$ for all $n,m\ge N$. Fix $n\ge N$ and let $u:W^{\circ}_{d}\to\mathbb{C}$ be the map $u(w)=x_{n}(w)-x(w)$. For $m\in\mathbb{N}$ and $w\in W^{\circ}_{d}$, $(x_{n}-x_{m})(w)=x_{n}(w)-x_{m}(w)$ by (1a), and $|x_{n}(w)-x_{m}(w)|^{2}=|x_{m}(w)-x_{n}(w)|^{2}$ by (0a); by (0d)(iii) with $b=x_{n}(w)$, this converges as $m\to\infty$ to $|x(w)-x_{n}(w)|^{2}=|u(w)|^{2}$, so $c_{w}|(x_{n}-x_{m})(w)|^{2}$ converges to $c_{w}|u(w)|^{2}$ by claim 3 of Arithmetic of Limits of Real Sequences. Fix $K\in\mathbb{N}$. By (0e)(ii) on each $W^{\circ}_{d,k}$, then Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §limit (with $N=j=K$) and claim 1 of Arithmetic of Limits of Real Sequences, the sequence $(T_{K}(x_{n}-x_{m}))_{m}$ converges to $T_{K}(u)$. For $m\ge N$, (1b) and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field give $T_{K}(x_{n}-x_{m})\le\lVert x_{n}-x_{m}\rVert_{d}^{2}<(\varepsilon/2)^{2}$, so $T_{K}(u)\le(\varepsilon/2)^{2}$ by (0c). As $K$ was arbitrary and $0\le c_{\varnothing}|u(\varnothing)|^{2}$, the partial sums $\sum_{k=1}^{K}U_{k}$ of the nonnegative block sums $U_{k}$ of $w\mapsto c_{w}|u(w)|^{2}$ satisfy $\sum_{k=1}^{K}U_{k}\le(\varepsilon/2)^{2}-|u(\varnothing)|^{2}$ for all $K$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion, $\sum_{k=1}^{\infty}U_{k}$ converges, and it equals the supremum of these partial sums, which is at most the upper bound $(\varepsilon/2)^{2}-|u(\varnothing)|^{2}$. Hence $u\in E_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space) and $\lVert u\rVert_{d}^{2}\le(\varepsilon/2)^{2}$, so $\lVert u\rVert_{d}\le\varepsilon/2<\varepsilon$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

(4c) Conclusion. Apply (4b) with $\varepsilon=1$ and $n=N$: the map $u$ lies in $E_{d}$, and $x=x_{N}+(-1)u$ pointwise, since $x_{N}(w)-(x_{N}(w)-x(w))=x(w)$; so $x\in E_{d}$ by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space. Now for any $\varepsilon>0$, with $N$ as in (4b), every $n\ge N$ satisfies $(x_{n}-x)(w)=x_{n}(w)-x(w)=u(w)$ by (1a), so $d_{E}(x_{n},x)=\lVert u\rVert_{d}<\varepsilon$. Thus $(x_{n})$ converges to $x\in E_{d}$ (Convergent Sequence in a Metric Space), and $E_{d}$ is a real Hilbert space. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §hilbert.

Step 5 (Clause 4: the metric). For $\lambda\in\mathcal{L}_{d}$, $\iota_{d}(\lambda)\in E_{d}$ is the restriction of $\lambda$ to $W^{\circ}_{d}$ (The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding), and by The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance and Step 3, $d_{\mathcal{L}}(\mu,\nu)=\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d}=d_{E}(\iota_{d}(\mu),\iota_{d}(\nu))$ for $\mu,\nu\in\mathcal{L}_{d}$, a real number. Let $\mu,\nu,\kappa\in\mathcal{L}_{d}$. Since $d_{E}$ is a metric on $E_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric, conditions 1, 3 and 4 of Metric Space for $d_{\mathcal{L}}$ at $(\mu,\nu,\kappa)$ are those for $d_{E}$ at $(\iota_{d}(\mu),\iota_{d}(\nu),\iota_{d}(\kappa))$. For condition 2: $d_{\mathcal{L}}(\mu,\nu)=0$ holds if and only if $\iota_{d}(\mu)=\iota_{d}(\nu)$, by condition 2 for $d_{E}$. If $\mu=\nu$ this holds. Conversely, if $\iota_{d}(\mu)=\iota_{d}(\nu)$, then $\mu(u)=\nu(u)$ for every $u\in W^{\circ}_{d}$, so $\mu=\nu$ by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §determined. Hence $d_{\mathcal{L}}$ is a metric on $\mathcal{L}_{d}$, proving The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric.

Step 6 (Clause 5: convergence is convergence on words). Let $(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{L}_{d}$ and $\lambda\in\mathcal{L}_{d}$, and put $\delta_{n}=\iota_{d}(\lambda_{n})-\iota_{d}(\lambda)\in E_{d}$, so that $\delta_{n}(w)=\lambda_{n}(w)-\lambda(w)$ for $w\in W^{\circ}_{d}$ by (1a) and $d_{\mathcal{L}}(\lambda_{n},\lambda)=\lVert\delta_{n}\rVert_{d}$.

($\Rightarrow$) Suppose $(\lambda_{n})$ converges to $\lambda$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$, and let $w\in W_{2d}$. By Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §reduction there is $u\in W^{\circ}_{d}$ with $\mu(w)=\mu(u)$ for every $\mu\in\mathcal{L}_{d}$. Let $\varepsilon>0$ and choose $N$ with $d_{\mathcal{L}}(\lambda_{n},\lambda)<\eta_{u}(\varepsilon)$ for $n\ge N$ (the number $\eta_{u}(\varepsilon)>0$ of (1c)). By (1c), for $n\ge N$, $|\lambda_{n}(w)-\lambda(w)|=|\lambda_{n}(u)-\lambda(u)|=|\delta_{n}(u)|<\varepsilon$. So $(\lambda_{n}(w))$ converges to $\lambda(w)$ in $(\mathbb{C},d_{\mathbb{C}})$.

($\Leftarrow$) Suppose $(\lambda_{n}(w))$ converges to $\lambda(w)$ in $(\mathbb{C},d_{\mathbb{C}})$ for every $w\in W_{2d}$. Let $D_{n,k}$ be the $k$-th block sum of $w\mapsto c_{w}|\delta_{n}(w)|^{2}$.

(6a) Uniform bound. For $w\in W^{\circ}_{d}$, by (0a) and Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound, $|\delta_{n}(w)|\le|\lambda_{n}(w)|+|\lambda(w)|\le2$, so $|\delta_{n}(w)|^{2}\le4$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence for $w\in W^{\circ}_{d,k}$, $0\le c_{w}|\delta_{n}(w)|^{2}\le4\theta^{k}$, and Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count (with $B=W^{\circ}_{d,k}$ and $M=4\theta^{k}$) gives $D_{n,k}\le4\theta^{k}(2d)^{k}=4r^{k}$, the last equality as computed in The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding. Moreover $\delta_{n}(\varnothing)=1-1=0$ by Laws of d-Tuples of Unitaries §normalised.

(6b) Choice of $K$. Let $\varepsilon>0$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $1-r$ is positive, $\sum_{k=1}^{\infty}r^{k}$ converges, $(r^{k})_{k}$ converges to $0$, and $\sum_{k=1}^{\infty}r^{k}-\sum_{k=1}^{K}r^{k}=r^{K+1}/(1-r)$ for every $K$. Put $\eta=(1-r)\varepsilon^{2}/8>0$ and first choose $K\in\mathbb{N}$ with $r^{k}=|r^{k}-0|<\eta$ for all $k\ge K$ (the powers being nonnegative by claim 5 of Properties of Natural Number Powers in a Field); since $K+1=S(K)$ (Natural Numbers) and $K<S(K)$, hence $K\le S(K)$ (claims 5 and 1 of Properties of the Order on the Natural Numbers), we get $r^{K+1}<\eta$. Every $r^{k}$ is positive (claims 5 and 4 of Properties of Natural Number Powers in a Field). Apply Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound with $\mu_{k}=r^{k}$, $M=4$ and $w_{k}=D_{n,k}\,(r^{k})^{-1}$, which satisfies $0\le w_{k}\le4$ by (6a); since $\mu_{k}w_{k}=D_{n,k}$, it gives, for every $n$,

$$0\le\sum_{k=1}^{\infty}D_{n,k}-\sum_{k=1}^{K}D_{n,k}\le4\,\frac{r^{K+1}}{1-r}<\frac{4\eta}{1-r}=\frac{\varepsilon^{2}}{2}.$$

(6c) Choice of $N$. Let $k\in[K]$ and $w\in W^{\circ}_{d,k}$. By hypothesis $(\lambda_{n}(w))_{n}$ converges to $\lambda(w)$, so by (0d)(iii) with $b=\lambda(w)$ the sequence $(|\delta_{n}(w)|^{2})_{n}$ converges to $|\lambda(w)-\lambda(w)|^{2}=0$, and $(c_{w}|\delta_{n}(w)|^{2})_{n}$ converges to $0$ by claim 3 of Arithmetic of Limits of Real Sequences. By (0e)(ii) and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, $(D_{n,k})_{n}$ converges to $0$; by Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §limit (with $N=j=K$) and Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §nonnegative, $(\sum_{k=1}^{K}D_{n,k})_{n}$ converges to $0$. Having fixed $K$ in (6b), now choose $N$ with $\sum_{k=1}^{K}D_{n,k}<\varepsilon^{2}/2$ for all $n\ge N$ (these sums being nonnegative).

(6d) Conclusion. For $n\ge N$, by Step 1, (6a), (6b) and (6c),

$$\lVert\delta_{n}\rVert_{d}^{2}=|\delta_{n}(\varnothing)|^{2}+\sum_{k=1}^{K}D_{n,k}+\Bigl(\sum_{k=1}^{\infty}D_{n,k}-\sum_{k=1}^{K}D_{n,k}\Bigr)<0+\frac{\varepsilon^{2}}{2}+\frac{\varepsilon^{2}}{2}=\varepsilon^{2},$$

so $d_{\mathcal{L}}(\lambda_{n},\lambda)=\lVert\delta_{n}\rVert_{d}<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence $(\lambda_{n})$ converges to $\lambda$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §convergence.

Step 7 (Clause 6: sequential compactness). Let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{L}_{d}$.

(7a) A subsequence converging on every word. The set $W_{2d}$ contains $\varnothing$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words), so it is nonempty, and it is countable by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §countable; by Countable Set there is a sequence $(w_{k})_{k\in\mathbb{N}}$ in $W_{2d}$ such that every $w\in W_{2d}$ equals $w_{k}$ for some $k$. For $m,k\in\mathbb{N}$ put $a_{m,k}=\operatorname{Re}\lambda_{m}(w_{k})$; by (0a) and Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound, $|a_{m,k}|\le|\lambda_{m}(w_{k})|\le1$. By The Diagonal Subsequence Lemma for Bounded Real Arrays (with $R_{k}=1$) there is a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ such that $(a_{n_{j},k})_{j}$ converges for every $k$. Likewise $b_{j,k}=\operatorname{Im}\lambda_{n_{j}}(w_{k})$ satisfies $|b_{j,k}|\le1$, and The Diagonal Subsequence Lemma for Bounded Real Arrays gives a strictly increasing $(l_{i})_{i\in\mathbb{N}}$ such that $(b_{l_{i},k})_{i}$ converges for every $k$. Put $m_{i}=n_{l_{i}}$. By claims 2 and 3 of A Subsequence of a Subsequence is a Subsequence, $(m_{i})_{i}$ is strictly increasing, $(\lambda_{m_{i}})_{i}$ is a subsequence of $(\lambda_{m})_{m}$, and $(a_{m_{i},k})_{i}$ is the subsequence of $(a_{n_{j},k})_{j}$ determined by $(l_{i})$, hence convergent by (0b). So for every $k$ the sequences $(\operatorname{Re}\lambda_{m_{i}}(w_{k}))_{i}=(a_{m_{i},k})_{i}$ and $(\operatorname{Im}\lambda_{m_{i}}(w_{k}))_{i}=(b_{l_{i},k})_{i}$ converge; call their limits $\alpha_{k}$ and $\beta_{k}$. Define $\lambda:W_{2d}\to\mathbb{C}$ by $\lambda(w)=\alpha_{k}+\beta_{k}i$ for any $k$ with $w_{k}=w$; this does not depend on $k$, because if $w_{k}=w_{k'}$ the sequences defining $\alpha_{k},\alpha_{k'}$ coincide, as do those defining $\beta_{k},\beta_{k'}$, and real limits are unique (0b). By (0a), $\operatorname{Re}\lambda(w)=\alpha_{k}$ and $\operatorname{Im}\lambda(w)=\beta_{k}$, so by (0d)(i)

$$(\ast)\qquad (\lambda_{m_{i}}(w))_{i}\ \text{converges to}\ \lambda(w)\ \text{in}\ (\mathbb{C},d_{\mathbb{C}})\ \text{for every}\ w\in W_{2d}.$$

(7b) The limit is a unitary law. We check the conditions of Laws of d-Tuples of Unitaries §law. Laws of d-Tuples of Unitaries §normalised: $\lambda_{m_{i}}(\varnothing)=1$ for all $i$, so by $(\ast)$ and (0d)(ii) (a constant sequence converges to its value, and limits are unique) $\lambda(\varnothing)=1$. Laws of d-Tuples of Unitaries §cancellation and Laws of d-Tuples of Unitaries §cyclic: for $u,v\in W_{2d}$ and $l\in[2d]$, the sequences $(\lambda_{m_{i}}(u\,l\,l^{-1}\,v))_{i}$ and $(\lambda_{m_{i}}(uv))_{i}$ are equal, so their limits $\lambda(u\,l\,l^{-1}\,v)$ and $\lambda(uv)$ (by $(\ast)$) are equal by (0d)(ii); in the same way $\lambda(uv)=\lambda(vu)$. Laws of d-Tuples of Unitaries §positive: let $F\subseteq W_{2d}$ be nonempty and finite, and put $K_{i}(v,v')=\lambda_{m_{i}}(v^{*}v')$ and $K(v,v')=\lambda(v^{*}v')$ for $v,v'\in F$; by $(\ast)$, $(K_{i}(v,v'))_{i}$ converges to $K(v,v')$. By Positive Semidefinite Kernel on a Finite Set §kernel applied to the positive semidefinite kernel $K_{i}$, $K_{i}(v',v)=\overline{K_{i}(v,v')}$ for all $i$; the left side converges to $K(v',v)$ and the right side to $\overline{K(v,v')}$ by (0d)(iv), so $K(v',v)=\overline{K(v,v')}$ by (0d)(ii). Let $z:F\to\mathbb{C}$. For $(u,v)\in F\times F$, the sequence $(\overline{z(u)}\,z(v)\,K_{i}(u,v))_{i}$ converges to $\overline{z(u)}\,z(v)\,K(u,v)$ by (0d)(iv) with $\zeta=\overline{z(u)}\,z(v)$; since $F\times F$ is nonempty and finite (Positive Semidefinite Kernel on a Finite Set), (0e)(iii) shows that $(Q_{K_{i}}(z))_{i}$ converges to $Q_{K}(z)$. Each $Q_{K_{i}}(z)$ is real and nonnegative, so $\operatorname{Im}Q_{K_{i}}(z)=0$ and $\operatorname{Re}Q_{K_{i}}(z)=Q_{K_{i}}(z)\ge0$ by (0a). By (0d)(i), $(\operatorname{Im}Q_{K_{i}}(z))_{i}$ converges to $\operatorname{Im}Q_{K}(z)$, which is therefore $0$ by (0b); and $(\operatorname{Re}Q_{K_{i}}(z))_{i}$ converges to $\operatorname{Re}Q_{K}(z)$, which is nonnegative by claim 1 of Order Properties of Limits of Real Sequences, comparing with the constant sequence $0$. Hence $Q_{K}(z)=\operatorname{Re}Q_{K}(z)+0i$ is real and nonnegative, and $K$ is a positive semidefinite kernel on $F$. Thus $\lambda\in\mathcal{L}_{d}$ (Laws of d-Tuples of Unitaries §space).

(7c) Conclusion. By $(\ast)$ and the implication ($\Leftarrow$) of Step 6, applied to the sequence $(\lambda_{m_{i}})_{i\in\mathbb{N}}$ in $\mathcal{L}_{d}$ and to $\lambda\in\mathcal{L}_{d}$, the subsequence $(\lambda_{m_{i}})_{i}$ of $(\lambda_{m})_{m}$ converges to $\lambda$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sequential.

Step 8 (Clause 7: compactness). By Step 5, $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space; let $\mathcal{T}$ be the collection of subsets of $\mathcal{L}_{d}$ open in it, a topology by Metric Open Sets Form a Topology. By Step 7, the subset $\mathcal{L}_{d}$ of $\mathcal{L}_{d}$ is sequentially compact in $(\mathcal{L}_{d},d_{\mathcal{L}})$: every sequence with terms in $\mathcal{L}_{d}$ has a subsequence, given by a strictly increasing sequence of indices, converging to a point of $\mathcal{L}_{d}$. By A Sequentially Compact Subset of a Metric Space is Compact, $\mathcal{L}_{d}$ is compact in $(\mathcal{L}_{d},\mathcal{T})$, that is (Compact Topological Space and Compact Subset), compact as a topological space with the subspace topology $\{\mathcal{L}_{d}\cap U:U\in\mathcal{T}\}$. Since every $U\in\mathcal{T}$ is a subset of $\mathcal{L}_{d}$, $\mathcal{L}_{d}\cap U=U$, so this subspace topology is $\mathcal{T}$ itself, and $(\mathcal{L}_{d},\mathcal{T})$ is compact. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §compact.

Step 9 (Clause 8: length-weighted differences). Let $\mu,\nu\in\mathcal{L}_{d}$ and $\delta=\iota_{d}(\mu)-\iota_{d}(\nu)\in E_{d}$. By (1a) and The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding, $\delta(w)=\mu(w)-\nu(w)$ for every $w\in W^{\circ}_{d}$, which is the first assertion. As in (6a), $|\delta(w)|^{2}\le4$ for every $w\in W^{\circ}_{d}$, and $\delta(\varnothing)=0$.

(9a) An elementary bound. For every $k\in\mathbb{N}$, $k\le2^{k}$ in $\mathbb{R}$. By claims 2 and 5 of Properties of Natural Number Powers in a Field, $1=1^{k}\le2^{k}$ for every $k$, as $0\le1\le2$. For $k=1$: $1\le2=2^{1}$, where $2^{1}=2$ by claim 1 of that lemma, and $1\le2$ because $2=1+1$ (claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field) and $0<1$ (claim 6 of Elementary Order Arithmetic in an Ordered Field), so $1=0+1<1+1$ by claim 1 of that lemma. If $k\le2^{k}$, then, since $S(k)=k+1$ (Natural Numbers), the image of $S(k)$ is $k+1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $k+1\le2^{k}+2^{k}=2^{k}\cdot2=2^{S(k)}$ by claim 1 of Properties of Natural Number Powers in a Field. By Principle of Induction for the Natural Numbers the bound holds for every $k$.

(9b) Summability. Let $X_{k}$ and $Y_{k}$ be the $k$-th block sums of $w\mapsto c_{w}|\delta(w)|^{2}$ and of $w\mapsto c_{w}|w|\,|\delta(w)|^{2}$; the terms of both are nonnegative (Step 1 and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sobolev). For $w\in W^{\circ}_{d,k}$ the latter term is $k\,\theta^{k}|\delta(w)|^{2}$, which lies between $0$ and $4k\theta^{k}$; by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count (with $M=4k\theta^{k}$), (9a), The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §embedding and claims 5 and 3 of Properties of Natural Number Powers in a Field,

$$0\le Y_{k}\le4k\,\theta^{k}(2d)^{k}=4k\,r^{k}\le4\cdot2^{k}r^{k}=4(2r)^{k}.$$

Since $0\le2r=\tfrac{1}{1536}<1$, the series $\sum_{k=1}^{\infty}(2r)^{k}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, so $\sum_{k=1}^{\infty}4(2r)^{k}$ converges by Elementary Properties of Series of Real Numbers §linearity, and $\sum_{k=1}^{\infty}Y_{k}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison. Hence $w\mapsto c_{w}|w|\,|\delta(w)|^{2}$ is summable (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums), and since $\delta\in E_{d}$, $\lVert\delta\rVert_{d,1}$ is defined by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sobolev, with $\lVert\delta\rVert_{d,1}^{2}=c_{\varnothing}\,|\varnothing|\,|\delta(\varnothing)|^{2}+\sum_{k=1}^{\infty}Y_{k}=\sum_{k=1}^{\infty}Y_{k}$, because $|\varnothing|=0$.

(9c) Comparison. By claim 4 of Properties of a Sum over a Finite Index Set, $Y_{k}=kX_{k}$. As $0\le X_{k}$ and $1\le k$ (claim 2 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), $Y_{k}-X_{k}=(k-1)X_{k}\ge0$, so $X_{k}\le Y_{k}$ for every $k$, and Elementary Properties of Series of Real Numbers §order gives $\sum_{k=1}^{\infty}X_{k}\le\sum_{k=1}^{\infty}Y_{k}$. By Step 1 and $\delta(\varnothing)=0$,

$$\lVert\delta\rVert_{d}^{2}=|\delta(\varnothing)|^{2}+\sum_{k=1}^{\infty}X_{k}=\sum_{k=1}^{\infty}X_{k}\le\sum_{k=1}^{\infty}Y_{k}=\lVert\delta\rVert_{d,1}^{2}.$$

Both norms being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert\delta\rVert_{d}\le\lVert\delta\rVert_{d,1}$. This proves The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev.