The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's *Functional Analysis*) occasionally state and prove more general versions, for instance for $F$-spaces (i.e. metrizable by a complete translation-invariant metric, but not necessarily locally convex). Again, these qualify as classical results. It seems that the Banach property can be relaxed to a much weaker completeness condition, as long as the space is metrizable.

Even in the absence of metric, we still have a notion of completeness in general topological vector spaces (every Cauchy filter/net converges). However, the proof of the open mapping theorem for $F$-spaces relies on the Baire category theorem, so this really uses the complete metric in a non-trivial way. This leads me to the following question:

Question 1.Does the open mapping theorem generalize to arbitrary (not necessarily metrizable) complete topological vector spaces?

As the open mapping theorem comes in various forms (see e.g. Rudin Theorem 2.11), I may have to be a bit more specific. I am particularly interested in the following question:

Question 2.Are there complete topological vector spaces $X$ and $Y$ and a continuous linear surjection $T : X \to Y$ such that $T$ is not open?