Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by $$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$ For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.
Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by
$$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$
For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.