Are extensions of nuclear Fréchet spaces nuclear? - MathOverflow

Are extensions of nuclear Fréchet spaces nuclear?

2011-03-31T20:11:10Z

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

Answer by Ralf for Are extensions of nuclear Fréchet spaces nuclear?

2011-03-31T22:03:48Z

Thank you for the hint, Yemon. You are indeed right. I found the following paper which proves the lifting property that you mentioned: emis.de/journals/PM/55f1/pm55f107.ps.gz The splitting follows from exmaple 3 on p. 96. Thanks again :-)

Answer by Jochen Wengenroth for Are extensions of nuclear Fréchet spaces nuclear?

2012-02-01T09:02:22Z

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).