For all $f, g \in \mathcal{E}$, we define $$(f \mid g) = \int_{-\infty}^{+\infty} f(y) g(y) \,\mathrm{d}y.$$ We define $\gamma_\lambda : \mathbf{R} \rightarrow \mathbf{R}$ by $\gamma_\lambda(y) = \exp\left(-y^2/\lambda\right)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$. (a) Show that for all $f \in \mathcal{E}$, we have $(f \mid f) \geq 0$ with equality if and only if $f = 0$. (b) Show that for all $x \in \mathbf{R}$, $\tau_x\left(\gamma_\lambda\right)$ belongs to $\mathcal{E}$.
For all $f, g \in \mathcal{E}$, we define
$$(f \mid g) = \int_{-\infty}^{+\infty} f(y) g(y) \,\mathrm{d}y.$$
We define $\gamma_\lambda : \mathbf{R} \rightarrow \mathbf{R}$ by $\gamma_\lambda(y) = \exp\left(-y^2/\lambda\right)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$.
(a) Show that for all $f \in \mathcal{E}$, we have $(f \mid f) \geq 0$ with equality if and only if $f = 0$.
(b) Show that for all $x \in \mathbf{R}$, $\tau_x\left(\gamma_\lambda\right)$ belongs to $\mathcal{E}$.