The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, carried into $\mathbb{R}$ by the natural-number image, and let $d\in\mathbb{N}$. Let $W_{2d}$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, let $W^{\circ}_{d}$ be the set of cyclically reduced words and, for $k\in\mathbb{N}$, $W^{\circ}_{d,k}$ the set of those of length $k$, and let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples. Let $\theta_{d}$ and the weights $c_{w}$, the block sums, the gauge space $E_{d}$, with its pointwise sums, real multiples and differences $x-y=x+(-1)y$, and the gauge norm $\lVert\cdot\rVert_{d}$ 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. Let $\mathbb{C}$ be the field of complex numbers with modulus $|z|$, and let $(2d)^{k}$ and $\theta_{d}^{\,k}$ denote natural powers in $\mathbb{R}$.

1. (Embedding)¶ For $\lambda\in\mathcal{L}_{d}$, $\iota_{d}(\lambda):W^{\circ}_{d}\to\mathbb{C}$ is the restriction of $\lambda$ to $W^{\circ}_{d}$. It belongs to $E_{d}$: by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound one has $|\lambda(w)|\le1$, hence, since $0<c_{w}\le\theta_{d}^{\,k}$ by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights, $0\le c_{w}|\lambda(w)|^{2}\le\theta_{d}^{\,k}$ for $w\in W^{\circ}_{d,k}$, so by Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §count the $k$-th block sum of $w\mapsto c_{w}|\lambda(w)|^{2}$ is at most

$$\theta_{d}^{\,k}(2d)^{k}=(2d\,\theta_{d})^{k}=\bigl(\tfrac{1}{3072}\bigr)^{k},$$

the first equality by claim 3 of Properties of Natural Number Powers in a Field and the second because $2d\,\theta_{d}=\frac{2d}{6144\,d}=\frac{1}{3072}$; the geometric series of ratio $\tfrac{1}{3072}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, so the series of block sums converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison. The map $\lambda\mapsto\iota_{d}(\lambda)$ from $\mathcal{L}_{d}$ to $E_{d}$ is the embedding $\iota_{d}$.

2. (Gauge distance)¶ The gauge distance of $\mu,\nu\in\mathcal{L}_{d}$ is the real number

$$d_{\mathcal{L}}(\mu,\nu)=\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d},$$

where $\iota_{d}(\mu)-\iota_{d}(\nu)\in E_{d}$ by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space.