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

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers and $d\in\mathbb{N}$. Let $W_{2d}$, the lengths $|w|$, the set $W^{\circ}_{d}$ of cyclically reduced words and the sets $W^{\circ}_{d,k}$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words and Reduced and Cyclically Reduced Words in Unitary Letters, and let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples. Let the weights $c_{w}$, summable word functions and their sums, the gauge space $E_{d}$ with its norm $\lVert\cdot\rVert_{d}$ and inner product $\langle\cdot,\cdot\rangle_{d}$, and the length-weighted gauge $\lVert\cdot\rVert_{d,1}$ be as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge, and let the embedding $\iota_{d}$ and the gauge distance $d_{\mathcal{L}}$ be as in The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance. Let $\mathbb{C}$ be the field of complex numbers, with conjugation $z\mapsto\overline{z}$ and real part $\operatorname{Re}z$.

Then the following hold.

1. (A real inner product space)¶ $E_{d}$, with pointwise addition, pointwise multiplication by real scalars, the zero map as zero vector and the pairing $\langle\cdot,\cdot\rangle_{d}$, is a real inner product space whose norm is $\lVert\cdot\rVert_{d}$.

2. (A real Hilbert space)¶ $E_{d}$, with the structure of clause 1, is a real Hilbert space.

3. (Expansion of the inner product)¶ Let $x,y\in E_{d}$ and, for $k\in\mathbb{N}$, $P_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}\operatorname{Re}\bigl(\overline{x(w)}\,y(w)\bigr)$. Then the series $\sum_{k=1}^{\infty}P_{k}$ converges, and

$$\langle x,y\rangle_{d}=\operatorname{Re}\bigl(\overline{x(\varnothing)}\,y(\varnothing)\bigr)+\sum_{k=1}^{\infty}P_{k}.$$

4. (Metric)¶ $d_{\mathcal{L}}$ is a metric on $\mathcal{L}_{d}$.

5. (Convergence is convergence on words)¶ Let $(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{L}_{d}$ and $\lambda\in\mathcal{L}_{d}$. Then $(\lambda_{n})_{n\in\mathbb{N}}$ converges to $\lambda$ in the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ if and only if, for every $w\in W_{2d}$, the sequence $(\lambda_{n}(w))_{n\in\mathbb{N}}$ converges to $\lambda(w)$ in the metric space $(\mathbb{C},d_{\mathbb{C}})$, where $d_{\mathbb{C}}(z,z')=|z-z'|$ is the modulus metric of claim 9 of Properties of Complex Conjugation and Modulus.

6. (Sequential compactness)¶ Every sequence in $\mathcal{L}_{d}$ has a subsequence that converges in $(\mathcal{L}_{d},d_{\mathcal{L}})$ to an element of $\mathcal{L}_{d}$.

7. (Compactness)¶ $\mathcal{L}_{d}$ is compact in the topology induced by $d_{\mathcal{L}}$, that is, in the topological space formed by the subsets of $\mathcal{L}_{d}$ open in $(\mathcal{L}_{d},d_{\mathcal{L}})$, which is a topology by Metric Open Sets Form a Topology.

8. (Length-weighted differences)¶ For all $\mu,\nu\in\mathcal{L}_{d}$, the difference $\iota_{d}(\mu)-\iota_{d}(\nu)\in E_{d}$ (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space) is the map $w\mapsto\mu(w)-\nu(w)$ on $W^{\circ}_{d}$, the map $w\mapsto c_{w}\,|w|\,|\mu(w)-\nu(w)|^{2}$ on $W^{\circ}_{d}$ is summable, so that $\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}$ is defined, and

$$\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d}\le\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}.$$