Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$: $$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$ Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.
Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$:
$$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.