Exercise 9, § 2

Let $(E_\alpha)_{\alpha \in A}$ be a family of Hausdorff locally convex spaces contained in the same vector space, which is directed for the relation ⊃, such that, if $E_\beta \subset E_\alpha$, the topology of E_\beta is finer than the topology on E_\beta induced by that of E_\alpha. Let E be the intersection of the E_\alpha, endowed with a topology which is the supremum of the topologies on E induced by those on the E_\alpha. Show that if each E_\alpha is semi-reflexive, then E is semi-reflexive (consider an ultrafilter on a bounded subset of E).