Exercise 15, § 4

Let I be an infinite non-enumerable set. Let $\mathfrak{F}(I)$ be the family of finite subsets of I and E the direct sum space $\mathbf{R}^{(I)}$. For every $J \in \mathfrak{F}(I)$, let $F_J$ be the subspace $\mathbf{R}^J$ of E, the product of the factors whose indices belong to J, with the product topology; let $g_J$ be the canonical injection of F_j in E. Show that there exists a topology $\mathscr{T}_0$ on E that is compatible with the vector space structure of E, which make the $g_j$ continuous and which is strictly finer than the finest locally convex topology $\mathscr{T}$ which makes the $g_j$ continuous. (Note that the set V of the $x = (\xi_\lambda)_{\lambda \in I}$ in E such that $\sum_{\lambda \in I} |\xi_\lambda|^{1/2} \leq 1$ is a neighbourhood of 0 for a topology compatible with the vector space structure of E, not containing any absorbent, symmetric, convex set.)