Exercise 9, § 3

a) Let $E$ be a separated semi-barrelled space, $M$ a closed vector subspace of $E$, $E'$ the dual of $E$. Now prove that $E/M$ is semi-barrelled and that the strong topology $\beta(M^\circ, E/M)$ is identical with the topology induced on $M^\circ$ by the strong topology $\beta(E', E)$. (Note that it is sufficient to prove that a sequence $(x'_n)$ in $M^\circ$ which converges to 0 for $\beta(E', E)$ is bounded for $\beta(M^\circ, E/M)$.) Deduce that if $E$ is a space (DF), the same is true of $E/M$ (cf. IV, p. 63, exerc. 8). b) Let $E$ be a separated locally convex space, $M$ a vector subspace (not necessarily closed) of $E$. Prove that if $M$ is a semi-barrelled space, the strong topology $\beta(E'/M^\circ, M)$ is identical with the quotient topology by $M^\circ$ of the strong topology $\beta(E', E)$. (Reason as in a).) c) Prove that a separated and quasi-complete semi-barrelled space $E$ is complete (use b) applied to $E$ and $\hat{E}$). In particular a semi-reflexive semi-barrelled space is complete (cf. IV, p. 53, exerc. 6). d) Prove that the completion of a separated semi-barrelled space (resp. a space (DF)) is semi-barrelled (resp. a space (DF)). e) Let $(E_n)$ be a sequence of semi-barrelled spaces (resp. of spaces (DF)), $E$ a vector space, and for each $n$, $f_n$ a linear mapping of $E_n$ into $E$. Suppose that $E$ is the union of the $f_n(E_n)$; prove that, for the finest locally convex topology making the $f_n$ continuous, $E$ is semi-barrelled (resp. a space (DF)) (examine first the case where $E$ is topological direct sum of the $E_n$). If the $E_n$ are semi-reflexive (resp. reflexive) and if $E$ is separated, $E$ is semi-reflexive (resp. reflexive).