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.


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.


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?


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?


Thanks a lot.



Example of noncomplete quotient of complete lcs mod closed subspace
A counterexample for both the first and third questions can be found in [*Counterexamples in Topological Vector Spaces*](http://rads.stackoverflow.com/amzn/click/354011565X) by Khaleelulla (p. 108).


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.


