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This question already has an answer here:

The open maping theorem between banach spaces says.

Let $T:X\to Y$ be a linear,continuous and surjective map between the banach spaces $X,Y$ then $T$ is an open map.

I need examples to show that the completeness assumption are important even in the presence of completeness of the other space. Thanks!

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marked as duplicate by Norbert, Stefan Hamcke, Davide Giraudo, user1551, Lord_Farin Oct 11 '13 at 21:15

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ See this mathoverflow post for the harder part. Also, this post here. $\endgroup$ – David Mitra Oct 11 '13 at 15:45