Can anyone please suggest where to get the proof of the following generalized version of open map theorem: Let $T$ be a bounded linear map from a Banach space $X$ into a normed space $Y$. If the image $T(X)$ is non-meager in $Y$ then $T$ is surjective and an open mapping. Moreover $Y$ is complete.

Are there any other generalized versions also for the open map theorem?


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