Exercise 9, § 3

a)** Let E be a Hausdorff semi-barrelled space, M a closed vector subspace of E and $E'$ the dual of E. Show 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 enough 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 (DF) space, then so is $E/M$ ($cf.$ IV, p. 63, exerc. 8).

b) Let E be a locally convex Hausdorff space, M a (not-necessarily closed) vector subspace of E. Show that if M is a semi-barrelled space, then the strong topology $\beta(E'/M^\circ, M)$ is identical with the quotient topology by $M^\circ$ of the strong topology $\beta(E', E)$. (Argue as in a.) c) Show that a Hausdorff and quasi-complete semi-barrelled space E is complete (use b) applied to E and $\hat{E}$). In particular, a semi-barrelled, semi-reflexive space is complete (cf. IV, p. 52, exerc. 6). d) Show that the completion of a semi-barrelled (resp. (DF)) Hausdorff space is semi-barrelled (resp. a (DF) space). e) Let $(E_n)$ be a sequence of semi-barrelled (resp. (DF)) spaces, E a vector space, and for each n, let $f_n$ be a linear mapping from $E_n$ into E. Suppose that E is the union of the $f_n(E_n)$; show that, E is semi-barrelled (resp. a (DF) space) for the finest locally convex topology for which all the $f_n$ are continuous (first examine the case where E is the topological direct sum of the $E_n$). If the $E_n$ are semi-reflexive (resp. reflexive) and if E is Hausdorff, then E is semi-reflexive (resp. reflexive).