Are extensions of nuclear Fréchet spaces nuclear? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:41:47Z http://mathoverflow.net/feeds/question/60230 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60230/are-extensions-of-nuclear-frechet-spaces-nuclear Are extensions of nuclear Fréchet spaces nuclear? Ralf 2011-03-31T20:11:10Z 2012-02-02T17:22:39Z <p>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:</p> <p>$0 \rightarrow V_1 \rightarrow V_2 \rightarrow V_3 \rightarrow 0$.</p> <p>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. </p> <p>Cheers,</p> <p>Ralf</p> http://mathoverflow.net/questions/60230/are-extensions-of-nuclear-frechet-spaces-nuclear/60239#60239 Answer by Ralf for Are extensions of nuclear Fréchet spaces nuclear? Ralf 2011-03-31T22:03:48Z 2011-03-31T22:03:48Z <p>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 :-)</p> http://mathoverflow.net/questions/60230/are-extensions-of-nuclear-frechet-spaces-nuclear/87212#87212 Answer by Jochen Wengenroth for Are extensions of nuclear Fréchet spaces nuclear? Jochen Wengenroth 2012-02-01T09:02:22Z 2012-02-01T09:02:22Z <p>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).</p> <p>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).</p>