Exercise 1, § 6

An exhaustion of a Hausdorff locally convex space is given by a sieve $C = (C_n, p_n)_{n \geq 0}$ (GT, IX, § 6, No. 5, def. 8) and, for every $n \geq 0$, a mapping $\rho_n$ from $C_n$ into the set of all convex and balanced subsets of $E$ having the following properties: E1) $E$ is the union of the $\phi_0(c)$ where $c$ ranges over $C_0$; E2) for every $n$ and every $c \in C_n$, $\phi_n(c)$ is the union of the $\phi_{n+1}(c')$ where $c'$ ranges over $p_n^{-1}(c)$; E3) for every sequence $(c_k)_{k \geq 0}$ such that $c_k \in C_k$ and $c_k = p_k(c_{k+1})$ for all $k \geq 0$, there exists a sequence $(\phi_k)$ of numbers $> 0$ such that, for every sequence $(x_k)$ of points of $E$ such that $x_k \in \phi_k(c_k)$ and every sequence $(\lambda_k)$ of real numbers satisfying $0 \leq \lambda_k \leq \rho_k$ for all $k$, the series $\sum_{k=0}^\infty \lambda_k x_k$ is convergent in $E$. a) Under the above hypotheses, show that if in addition the $\phi_n(c)$ are closed for $c \in C_n$, then we can assume that the $\rho_k$ have been so chosen that we have $\sum_{k=m}^\infty \lambda_k x_k \in \phi_m(c_m)$ for all $m \geq 1$ (take the $\rho_k$ such that $\sum_{k=0}^\infty \rho_k \leq 1$). b) Suppose that we are given a sieve $C$ and sequence $(\phi_n)$ of mappings into the set of convex and balanced subsets of $E$ satisfying E1), E2) and the following condition: E3') for every sequence $(c_k)_{k \geq 0}$ such that $c_k = p_k(c_{k+1})$ for all $k \geq 0$, there exists a sequence $(\mu_k)$ of numbers $> 0$ such that, for every sequence $(x_k)$ of points of $E$ with $x_k \in \phi_k(c_k)$ for all $k$, the sequence of points $(\mu_k x_k)$ is contained in a convex, bounded balanced and semi-complete set in $E$. Show that then the condition E3) is also verified (take $\rho_k = 2^{-k} \mu_k$). A locally convex Hausdorff space is said to be exhaustible if there exists an exhaustion of $E$.