This question is related to one I posted before: https://mathoverflow.net/questions/117492/dfm-spaces-and-the-c-infty-topology
According to Dineen (in "Complex Analysis in Locally Convex Spaces",p.15) a (DFM)-space $(E,\tau)$ is a (DFS)-space if and only if $\tau = c^{\infty}$, i.e. if the given topology coincides with the Mackey-closure (= $c^{\infty}$-topology) on $(E,\tau)$. However, there is neither a proof nore a reference stated, and I cannot see how the facts that a $E$ carries the Mackey-closure-topology and is a (DFM)-space lead to the consequence that it is allready a (DFS)-space. If $E$ is a (DFM)-space then there is a sequence of Banach-spaces $E_{n}$ so that $E = \varinjlim E_{n}$ and which is regular -i.e every bounded set of $E$ is allready contained in some $E_{n}$. Since $E$ carries the Mackey-closure-topology, we have that $E= \varinjlim E_{n}$ in the category of topological spaces. Since $E$ is a Montel space, every bounded set is relatively compact in $E$. For showing that $E$ is a (DFS)-space one would to need to be able to conclude that every compact set of $E$ is allready compact in the Banach space topology of some step $E_{n}$ (with $n$ depending on the compact set). However, at this point I fail. Can somebody give me reference on the proof of this theorem or even complete the argument?
An interesting consequence of stated theorem is that if $E = \bigcup_{r \in (0,1)} E_{r}$ is the limit of a Banach scale, then $E$ is either metrizable or a (DFS)-space. A Banach scale is a net $(E_{r})$ of Banach spaces such that $E_{r}$ includes continuously into $E_{s}$ if $r>s$ which satisfies that $E = \bigcup E_{r}$ equipped with the finest topology such that all injections $\iota_{r} \colon E_{r} \to E$ are continuous is a locally convex space. This follows from a theorem of Kakol/Saxon (Montel (DF)-spaces, sequential (LM)- spaces and the strongest locally convex topology. J. Lond. Math. Soc., II. Ser., 66(2):388–406, 2002) which states that an inductive limit of metrizable locally convex spaces is sequential if and only if it is either metrizable or a (DFM)-space.
