Let $\lambda > 0$ be fixed. We use the notation $\mathcal{G}$, $\mathcal{H} = C(\mathcal{G})$, $\gamma_\lambda$, $\tau_x$, and $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$ as defined previously.
(a) Show that there exists $c_\lambda > 0$ such that for all $(x, x') \in \mathbf{R} \times \mathbf{R}$ we have $$\left(\tau_x(\gamma_\lambda) \mid \tau_{x'}(\gamma_\lambda)\right) = c_\lambda \gamma_{2\lambda}(x - x')$$ Hint: One may note that $\frac{1}{\lambda}\left((y-x)^2 + (y-x')^2\right) = \frac{2}{\lambda}\left(y - (x+x')/2\right)^2 + \frac{1}{2\lambda}(x'-x)^2$.
(b) Deduce that for all $x \in \mathbf{R}$ $$C\left(\tau_x(\gamma_\lambda)\right) = c_\lambda \tau_x(\gamma_{2\lambda})$$ and that $$\mathcal{H} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda}) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$