DF-algebras and DF-modules Recall Lemma 0.5.1 from the Helemskii's monograph "The homology of Banach and Topological Algebras":
$\textbf{Lemma}$ Let $\phi\colon X\to Y$ be an injective map between Banach spaces with dense range. If the dual map $\phi^*$ is surjective then so is $\phi$ itself.
My question is: what if $X,Y$ are complete DF-spaces? Is this lemma still true? The situation I am considering is even more specific. Namely, $X$ is a quotient of a complete DF-space (by a closed subspace) and $Y$ is a closed subspace of a complete DF-space.
The only thing I can see is that $\phi^*$ is a topological isomorphism $\textit{onto}$.
 A: Here is a counterexample: For every countable inductive limit $E=\lim E_n$
of Banach (or locally convex) spaces one has (either by definition or a simple construction) an algebraically exact sequence $0\to \bigoplus E_n \to \bigoplus E_n \to E\to 0$ where the map $\phi$ on the direct sum is
defined by $(e_n)_{n\in\mathbb N} \mapsto (e_n-j_{n-1}(e_{n-1}))_{n\in\mathbb N}$ (where $j_n:E_n\to E_{n+1}$ are the linking maps of the inductive spectrum and $j_0(e_0)=0$).
There are inductive limits $E$ of Banach spaces such that the limit topology is trivial (only the empty set and the whole space are open). In this case $X=Y=\bigoplus E_n$ are (very nice, in particular, barrelled) DF-spaces and $\phi$ is continuous with dense range (because the associated Hausdorff space of $E$ is $\lbrace 0\rbrace =\bigoplus E_n/\overline{\text{range}(\phi)}$. Then $\phi^*$ is an isomorphism
but $\phi$ is not surjective.
As mentioned in the comments, the answer to your question is positive if
$X$ is a reflexive DF-space or if $\phi$ is the transposed of an operator between Frechet spaces (in that case the closed range theorem or the arguments in my third comment apply). 
