The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge

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 $W^{\circ}_{d}$ be the set of cyclically reduced words and, for $k\in\mathbb{N}$, $W^{\circ}_{d,k}$ the set of those of length $k$, with lengths $|w|$. Let $\mathbb{C}$ be the field of complex numbers with modulus $|z|$. By Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k §finite every $W^{\circ}_{d,k}$ is nonempty and finite; sums over it are those of Sum over a Finite Index Set, and series and their convergence are those of Series of Real Numbers.

1. (Weights)¶ Put $\theta_{d}=\frac{1}{6144\,d}$. For $w\in W^{\circ}_{d}$ put $c_{w}=1$ if $w=\varnothing$, and

$$c_{w}=\frac{\theta_{d}^{\,k}}{(k+1)^{4}}$$

if $w$ has length $k\in\mathbb{N}$, where $\theta_{d}^{\,k}$ and $(k+1)^{4}$ are natural powers. Every $c_{w}$ is positive: $c_{\varnothing}=1$ is positive by Elementary Order Arithmetic in an Ordered Field, claim 6; for $w\ne\varnothing$, $\theta_{d}$ is positive, as the quotient of $1$ by the positive real number $6144\,d$, and $k+1$ is positive; natural powers of positive reals are nonnegative and nonzero by claims 5 and 4 of Properties of Natural Number Powers in a Field, hence positive, and a quotient of positive reals is positive. Moreover $c_{w}\le\theta_{d}^{\,k}$: since $1\le k+1$, one has $1=1^{4}\le(k+1)^{4}$ by claims 2 and 5 of Properties of Natural Number Powers in a Field, and dividing the positive number $\theta_{d}^{\,k}$ by a real number at least $1$ does not increase it.

2. (Summable word functions)¶ A map $a:W^{\circ}_{d}\to\mathbb{R}$ with $0\le a(w)$ for every $w\in W^{\circ}_{d}$ is summable if the series $\sum_{k=1}^{\infty}A_{k}$ converges, where $A_{k}=\sum_{w\in W^{\circ}_{d,k}}a(w)$ is its $k$-th block sum. Its sum is then the real number

$$\sum_{w\in W^{\circ}_{d}}a(w)=a(\varnothing)+\sum_{k=1}^{\infty}A_{k}.$$

3. (The gauge space)¶ $E_{d}$ is the set of maps $x:W^{\circ}_{d}\to\mathbb{C}$ for which $w\mapsto c_{w}\,|x(w)|^{2}$ is summable. For $x,y\in E_{d}$ and $s\in\mathbb{R}$, the pointwise sum $x+y$ and the pointwise multiple $sx$ belong to $E_{d}$. Indeed, let $w\in W^{\circ}_{d}$. By the triangle inequality, claim 7 of Properties of Complex Conjugation and Modulus, $|x(w)+y(w)|\le|x(w)|+|y(w)|$, both sides being nonnegative, so $|x(w)+y(w)|^{2}\le\bigl(|x(w)|+|y(w)|\bigr)^{2}$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and for real $a,b$ one has $(a+b)^{2}\le2a^{2}+2b^{2}$, since $2a^{2}+2b^{2}-(a+b)^{2}=(a-b)^{2}\ge0$. Hence $|x(w)+y(w)|^{2}\le2|x(w)|^{2}+2|y(w)|^{2}$. Moreover $|s\,x(w)|=|s|\,|x(w)|$ by claim 4 of Properties of Complex Conjugation and Modulus, and $|s|$ equals $s$ or $-s$ by claim 8 of that lemma, so $|s\,x(w)|^{2}=s^{2}|x(w)|^{2}$. Since every $c_{w}$ is positive by clause 1, the block sums of $c_{w}|x(w)+y(w)|^{2}$ and of $c_{w}|s\,x(w)|^{2}$ are at most $2X_{k}+2Y_{k}$ and equal to $s^{2}X_{k}$, where $X_{k}$ and $Y_{k}$ are the block sums for $x$ and $y$, and Elementary Properties of Series of Real Numbers §linearity together with Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison gives convergence. For $x,y\in E_{d}$ we write $x-y=x+(-1)y$, which belongs to $E_{d}$.

4. (The gauge norm)¶ Let $x\in E_{d}$. The sum $\sum_{w\in W^{\circ}_{d}}c_{w}|x(w)|^{2}$ is nonnegative: every term $c_{w}|x(w)|^{2}$ is nonnegative, since $c_{w}$ is positive by clause 1 and $0\le|x(w)|^{2}$; hence every block sum is nonnegative as a finite sum of nonnegative terms, by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative; the convergent series of these block sums therefore has a nonnegative sum by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates; and adding the nonnegative term $c_{\varnothing}|x(\varnothing)|^{2}$ keeps it nonnegative. The gauge norm $\lVert x\rVert_{d}$ is the nonnegative square root of this sum, given by Existence and Uniqueness of the Nonnegative Square Root.

5. (The gauge inner product)¶ For $x,y\in E_{d}$, with $x+y$ and $x-y$ in $E_{d}$ by clause 3, the gauge inner product is

$$\langle x,y\rangle_{d}=\tfrac14\bigl(\lVert x+y\rVert_{d}^{2}-\lVert x-y\rVert_{d}^{2}\bigr).$$

6. (The length-weighted gauge)¶ Let $x\in E_{d}$ be such that the map $w\mapsto c_{w}\,|w|\,|x(w)|^{2}$ on $W^{\circ}_{d}$ is summable; its values are nonnegative, since $c_{w}$ is positive, $0\le|x(w)|^{2}$, and $0\le|w|$, the length $|w|$ being $0$ or the image of a natural number, which is positive by The Real Numbers: Standing Notation and Background §numbers. Then $\lVert x\rVert_{d,1}$ denotes the nonnegative square root of its sum, which is nonnegative as in clause 4.