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.