Exercise 2, § 6

Let $E$ be a locally convex space which is a Baire space, $F$ a locally convex exhaustible space (exerc. 1), and $(C_n, p_n, \phi_n)$ an exhaustion of $F$. a) Let $u$ be a linear mapping from $E$ into $F$ and let $W$ be a convex, balanced and absorbent set in $F$. Show that there exists a sequence $(c_k)$ such that $c_k \in C_k$, $c_k = p_k(c_{k+1})$ for all $k \geq 0$, and a sequence $(m_k)$ of integers $> 0$, such that each of the sets $u^{-1}(\phi_k(c_k) \cap m_k W)$ (which is denoted by $M_k$) is not a thin set in $E$. Show that for every $\varepsilon > 0$, there exists a sequence $(v_k)$ of numbers $> 0$ such that if the sequence $(x_k)_{k \geq 1}$ of points of $E$ is such that $x_k \in v_k M_k$ for all $k \geq 1$, the serie $\sum_{k=1}^\infty u(x_k)$ converges in $F$ and that its sum belongs to $\varepsilon W$. b) Suppose, in addition, that $E$ is metrizable and that the graph of $u$ in $E \times F$ is closed. Show that for every $\varepsilon > 0$, we have $u^{-1}(W) \subset (1+\varepsilon) u^{-1}(W)$. (Observe that if $(U_k)$ is a countable fundamental system of neighbourhoods of $0$ in $E$, then for every $k$ there exists a convex balanced neighbourhood $V_k$ of $0$ in $E$ such that $V_k \subset U_k \cap v_k \overline{M}_k$. For every point $a \in u^{-1}(W)$, find a sequence $(x_k)_{k \geq 0}$ such that $x_0 \in u^{-1}(W)$, $x_k \in v_k M_k$ for $k \geq 1$ and $a - \sum_{j=0}^k x_j \in V_k$ for all $k \geq 1$, then apply a).)

c) Deduce from b) that if E is a metrizable Baire space, then every linear mapping from E into F whose graph is closed is continuous.