Exercise 3, § 3

Let $E$ be a topological vector space over $\mathbf{R}$ (resp. $\mathbf{C}$). Show that for every family $\mathfrak{S}$ of bounded subsets of $\mathbf{R}$ (resp. $\mathbf{C}$) none of which is the point 0, the space $\mathscr{L}_{\mathfrak{S}} (\mathbf{R} ; E)$ (resp. $\mathscr{L}_{\mathfrak{S}} (\mathbf{C} ; E)$) is canonically isomorphic to $E$. Deduce that for every integer $n > 0$ and every covering $\mathfrak{S}$ of $\mathbf{R}^n$ (resp. $\mathbf{C}^n$) by bounded subsets, $\mathscr{L}_{\mathfrak{S}} (\mathbf{R}^n ; E)$ (resp. $\mathscr{L}_{\mathfrak{S}} (\mathbf{C}^n ; E)$) is isomorphic to $E^n$.