Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, let $d\in\mathbb{N}$, let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples, and let the gauge space $E_{d}$ with its 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 and the embedding $\iota_{d}:\mathcal{L}_{d}\to E_{d}$ as in The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance; sums and real multiples in $E_{d}$ are pointwise, and $x-y=x+(-1)y$. Let $\lambda\in\mathcal{L}_{d}$ and $v\in E_{d}$.

1. (Inward tangent vectors)¶ $v$ is an inward tangent vector to $\mathcal{L}_{d}$ at $\lambda$ if there are a real $s_{0}>0$ and a map $\gamma$ from $\{s\in\mathbb{R}:0\le s\le s_{0}\}$ to $\mathcal{L}_{d}$ with $\gamma(0)=\lambda$ such that for every real $\varepsilon>0$ there is a real $\delta>0$ with

$$\Bigl\lVert\tfrac{1}{s}\bigl(\iota_{d}(\gamma(s))-\iota_{d}(\lambda)\bigr)-v\Bigr\rVert_{d}\le\varepsilon\qquad\text{for every real }s\text{ with }0<s\le s_{0}\text{ and }s<\delta .$$

The set of inward tangent vectors at $\lambda$ is written $T^{+}_{\lambda}$.

2. (Two-sided tangent vectors)¶ $v$ is a two-sided tangent vector to $\mathcal{L}_{d}$ at $\lambda$ if there are a real $s_{0}>0$ and a map $\gamma$ from $\{s\in\mathbb{R}:-s_{0}\le s\le s_{0}\}$ to $\mathcal{L}_{d}$ with $\gamma(0)=\lambda$ such that for every real $\varepsilon>0$ there is a real $\delta>0$ with

$$\Bigl\lVert\tfrac{1}{s}\bigl(\iota_{d}(\gamma(s))-\iota_{d}(\lambda)\bigr)-v\Bigr\rVert_{d}\le\varepsilon\qquad\text{for every real }s\text{ with }0<|s|\le s_{0}\text{ and }|s|<\delta .$$

The set of two-sided tangent vectors at $\lambda$ is written $T^{\pm}_{\lambda}$.