We use the notation $\mathcal{G}$, $\mathcal{H}$, $\gamma_{2\lambda}$, $\tau_x$, $C$, $D$ as defined previously. (a) Let $n \in \mathbf{N}_*$ and $(x_i)_{1 \leq i \leq n}$ a family of real numbers such that for all $i, j \in \llbracket 1,n \rrbracket$ we have $x_i \neq x_j$ when $i \neq j$. Show that the function $\sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda})$ is zero if and only if $\alpha_i = 0$ for all $1 \leq i \leq n$ (Hint: One may proceed by induction on $n$). (b) Deduce that there exists a unique linear map $D$ from $\mathcal{H}$ to $\mathcal{G}$ such that $D \circ C(g) = g$ for all $g \in \mathcal{G}$ and $C \circ D(h) = h$ for all $h \in \mathcal{H}$. (c) Show that for all $h \in \mathcal{H}$, we have for all $x \in \mathbf{R}$ that $h(x) = \left(\tau_x(\gamma_\lambda) \mid D(h)\right)$.
We use the notation $\mathcal{G}$, $\mathcal{H}$, $\gamma_{2\lambda}$, $\tau_x$, $C$, $D$ as defined previously.
(a) Let $n \in \mathbf{N}_*$ and $(x_i)_{1 \leq i \leq n}$ a family of real numbers such that for all $i, j \in \llbracket 1,n \rrbracket$ we have $x_i \neq x_j$ when $i \neq j$. Show that the function $\sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda})$ is zero if and only if $\alpha_i = 0$ for all $1 \leq i \leq n$ (Hint: One may proceed by induction on $n$).
(b) Deduce that there exists a unique linear map $D$ from $\mathcal{H}$ to $\mathcal{G}$ such that $D \circ C(g) = g$ for all $g \in \mathcal{G}$ and $C \circ D(h) = h$ for all $h \in \mathcal{H}$.
(c) Show that for all $h \in \mathcal{H}$, we have for all $x \in \mathbf{R}$ that $h(x) = \left(\tau_x(\gamma_\lambda) \mid D(h)\right)$.