# Banach open mapping theorem pairs + inheritability on closed subspace of the image space

Given two Frechet spaces $E$ and $F$, and a continuous linear surjection $T\colon E \to F$, then $T$ is an open map. Now if $H$ is a closed subspace of $F$, and $T\colon E \to H$ is a linear surjection, we can again conclude that the map is open since $H$ is again a Frechet space. One of the generalizations of the open mapping theorem is formulation for the pair Ptak/Barreled (e.g. Schaefer "Topological Vector spaces"), i.e. if $E$ is a Ptak space, and $F$ a barreled space, every linear continuous surjection from $E$ to $F$ is open. However, a closed subspace of a barreled space is i.g. not barreled, so if $T\colon E \to H$ is a linear surjection on $H$ and $H$ a closed subspace of $F$, we cannot apply the open mapping theorem since $H$ might fail to be barreled.

My question is: Is there a class of spaces which is more general than the class of Frechet-spaces, for which the validity of the open-mapping-theorem is inherited by closed subspaces (where this class of spaces appears as image space in the open mapping theorem)?

I am not an expert on this matter, so maybe this question is ill-posed.

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I am afraid that the answer is no. One of he most general open mapping theorems due to de Wilde allows very general domains (so-called webbed locally convex spaces) but the range space must be ultrabornological = arbitrary (neither countable nor injective) inductive limit of Banach spaces. The class of ultrabornological spaces is not stable w.r.t. closed subspaces.

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Thank you for your reply. I did not really expect a positive answer, but it would had been quite usefull I think, so I decided to post it anyways. –  Sebastian Jul 2 '12 at 10:10