TITLE: Are extensions of nuclear Fréchet spaces nuclear?
QUESTION [15 upvotes]: Consider the category of Fréchet spaces, the morphisms being
continuous linear maps with closed image. Suppose that we 
have a short exact sequence in that category:
$0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$.
Of course $V_1$ and $V_3$ are nuclear if $V_2$ is. I recently asked
myself if the converse might be true. I haven't found anything useful
in the standard literature (Treves, Schaefer) but that might
be just me being too ignorant to see the obvious. I'm grateful if someone could 
shed some light on this. 
Cheers,
Ralf

REPLY [8 votes]: You question was answered even for locally convex spaces by S. Dierolf and W. Roelcke
in proposition 3.8 of the article "On the three-space-problem for topological vector
spaces". Collect. Math. 32, p. 13-35 (1981).
The splitting theory for Frechet spaces is nowadays very well understood by results of D. Vogt and others. This can be found in my "Derived functors in functional analysis", Springer Lecture Notes in Mathematics 1810 (2003).