Exercise 7, § 1

a) Let E be an infinite dimensional Banach space and let $\mathscr{S}$ be the family of compact, convex and balanced subsets of E, which is a directed set for the relation $\subset$. Show that E is the inductive limit of the inductive system of the Banach spaces $E_A$, where A runs through $\mathscr{S}$. (Prove by contradiction that a neighbourhood V of 0 for the inductive limit topology of the topologies of $E_A$ contains a ball with center 0; for this, note that otherwise, there will exist a sequence $(x_n)$ of points of E such that $\|x_n\| \leq 1/n^2$, and which will not belong to V.) Deduce from this that there exist bounded subsets in E which are not contained in any $E_A$ for $A \subset \mathscr{S}$.

b) Let E be an infinite dimensional Banach space. On the vector space $\prod_{m=1}^\infty E_m$, where $E_m = E$ for each m, let $\mathscr{T}_n$ denote the topology obtained by taking the product of the Banach space topology on each $E_m$ for $m \leq n$, and for $m > n$, the finest locally convex topology on $E_m$; $F_n$ denotes the locally convex space $\prod_{m=1}^\infty E_m$ with the topology $\mathscr{T}_n$. Every identity map $F_n \to F_{n+1}$ is continuous; show that the inductive limit space of the inductive system $F_n$ is the space F obtained by assigning to $\prod_{m=1}^\infty E_m$ the topology which is the product of the Banach space topologies on each of the factors. Deduce from this that there are bounded subsets in F which are not bounded in any $F_n$.