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This question already has an answer here:

This question gave an example of a continuous $f: E \rightarrow F$ which is bijective but has noncontinuous inverse. In the example, neither $E$ nor $F$ was a Banach space, are there any examples where $E$ or $F$ is a Banach space (of course, none exist if both $E$ and $F$ are Banach, by an application of the open mapping theorem).

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marked as duplicate by Nate Eldredge, TMM, Chris Janjigian, user85798, user61527 Mar 13 '14 at 3:05

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    $\begingroup$ See this. $\endgroup$ – David Mitra Mar 12 '14 at 23:37
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Let $E=C([0,1],\Bbb R)$ (continuous maps frome $[0,1]$ to $\Bbb R$) with $\|.\|_{\infty}$ and $\|.\|_1$ and identical mape : $f \mapsto f$ frome $(E,\|.\|_{\infty})$ to $(E,\|.\|_{1})$

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  • $\begingroup$ But, are both Banach spaces? $\endgroup$ – Berci Mar 12 '14 at 23:47
  • $\begingroup$ i am editing with an other example. the question says $$E$ or $F$ Banach ... $\endgroup$ – Mohamed Mar 13 '14 at 0:14
  • $\begingroup$ It's well known that $(E, \|\cdot\|_\infty)$ is Banach and $(E, \|\cdot\|_1)$ is not. $\endgroup$ – Nate Eldredge Mar 13 '14 at 0:32

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