Example of noncomplete quotient of complete lcs mod closed subspace - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:08:06Z http://mathoverflow.net/feeds/question/57654 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57654/example-of-noncomplete-quotient-of-complete-lcs-mod-closed-subspace Example of noncomplete quotient of complete lcs mod closed subspace Stefan Waldmann 2011-03-07T12:08:12Z 2011-03-09T01:34:42Z <p>The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a Banach space the same statement holds true.</p> <p>Beyond the metrizable case this is no longer correct. So my first question is about a good counter-example, i.e. a complete locally convex space $V$ with a closed subspace $W$ such that $V / W$ is no longer complete.</p> <p>My second question is whether counter-examples necessarily arise beyond the metrizable case, i.e. does every complete lcs have a closed subspace with a non-complete quotient? In other words, does the above quotient property characterize Fréchet spaces?</p> <p>My third question is how the situation looks like for sequentially complete lcs with sequentially closed subspace. Are there any positive results/situations where the quotient is at least sequentially complete again?</p> <p>Thanks a lot.</p> http://mathoverflow.net/questions/57654/example-of-noncomplete-quotient-of-complete-lcs-mod-closed-subspace/57660#57660 Answer by Andrey Rekalo for Example of noncomplete quotient of complete lcs mod closed subspace Andrey Rekalo 2011-03-07T13:15:18Z 2011-03-09T01:34:42Z <p>A counterexample for both the first and third questions can be found in <a href="http://www.amazon.com/Counterexamples-Topological-Vector-Lecture-Mathematics/dp/354011565X/ref=pd_sxp_f_pt" rel="nofollow"><em>Counterexamples in Topological Vector Spaces</em></a> by Khaleelulla (p. 108).</p> <p>Let $W$ denote the space of all $\mathbb C$-valued sequences $(x_n)$ and $\Phi$ the space of finite sequences. Let $E=E_1\oplus E_2$ where $E_1$ is the topological countable direct sum of copies of $W$ and $E_2$ is the topological countable product of copies of $\Phi$. $E$ is a complete locally convex space but the quotient $E/M$ where $M=\{(u,u):\ u\in E_1\cap E_2 \}$ is not even sequentially complete.</p> http://mathoverflow.net/questions/57654/example-of-noncomplete-quotient-of-complete-lcs-mod-closed-subspace/57663#57663 Answer by Bill Johnson for Example of noncomplete quotient of complete lcs mod closed subspace Bill Johnson 2011-03-07T13:39:38Z 2011-03-07T13:39:38Z <p>For the second question, consider the direct sum $Z$ of infinitely many copies of the scalar field. Given any subspace $X$, any algebraic complement $Y$ to the subspace is also a topological complement to $X$, so the quotient $Z/X$ is linearly homeomorphic to $Y$, which is again a direct sum of copies of the scalar field.</p>