We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $\delta(\boldsymbol{x}, \boldsymbol{y}) = \inf\{ \|\boldsymbol{y} - g \cdot \boldsymbol{x}\| \mid g \in \operatorname{Dep}(\mathbb{R}^{d}) \}$.
[(a)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $$\|g \cdot \boldsymbol{y} - g \cdot \boldsymbol{x}\| = \|\boldsymbol{y} - \boldsymbol{x}\|.$$
[(b)] Deduce that $\delta(\boldsymbol{x}, \boldsymbol{y}) = \delta(\boldsymbol{y}, \boldsymbol{x})$.
[(c)] Show that for all $(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}) \in \mathscr{E}_{d}^{n}(\mathbb{R})^{3}$ and $(g, g^{\prime}) \in (\operatorname{Dep}(\mathbb{R}^{d}))^{2}$, we have $$\|\boldsymbol{z} - g \cdot \boldsymbol{x}\| \leqslant \|\boldsymbol{z} - (gg^{\prime}) \cdot \boldsymbol{y}\| + \|g^{\prime} \cdot \boldsymbol{y} - \boldsymbol{x}\|$$
We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $\delta(\boldsymbol{x}, \boldsymbol{y}) = \inf\{ \|\boldsymbol{y} - g \cdot \boldsymbol{x}\| \mid g \in \operatorname{Dep}(\mathbb{R}^{d}) \}$.
\begin{itemize}
\item[(a)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have
$$\|g \cdot \boldsymbol{y} - g \cdot \boldsymbol{x}\| = \|\boldsymbol{y} - \boldsymbol{x}\|.$$
\item[(b)] Deduce that $\delta(\boldsymbol{x}, \boldsymbol{y}) = \delta(\boldsymbol{y}, \boldsymbol{x})$.
\item[(c)] Show that for all $(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}) \in \mathscr{E}_{d}^{n}(\mathbb{R})^{3}$ and $(g, g^{\prime}) \in (\operatorname{Dep}(\mathbb{R}^{d}))^{2}$, we have
$$\|\boldsymbol{z} - g \cdot \boldsymbol{x}\| \leqslant \|\boldsymbol{z} - (gg^{\prime}) \cdot \boldsymbol{y}\| + \|g^{\prime} \cdot \boldsymbol{y} - \boldsymbol{x}\|$$
\item[(d)] Deduce that $\delta(\boldsymbol{x}, \boldsymbol{z}) \leqslant \delta(\boldsymbol{x}, \boldsymbol{y}) + \delta(\boldsymbol{y}, \boldsymbol{z})$.
\end{itemize}