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Prove that a bounded linear bijection between two Banach spaces is a homeomorphism.

I need to show that both the operator and the inverse are continuous. The bounded linear operator is continuous, but I'm not sure how to show that the inverse is continuous. I know I need to use the open mapping at some point but not sure how to set up the problem. Can someone help!

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  • $\begingroup$ Use the topological definition of continuity: a map is continuous if and only of the preimage of any open set is open. Apply this to the inverse operator, and use the fact that the operator is bijective and open. $\endgroup$ – Gyu Eun Lee Dec 22 '16 at 14:06
  • $\begingroup$ It is important that they are Banach spaces and not merely normed linear spaces, I have seen an example of a continuous linear bijection $f:X\to X$ where $X$ is a normed space with an incomplete norm, such that $f^{-1}$ is not continuous. $\endgroup$ – DanielWainfleet Dec 28 '16 at 10:15
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This is a straightforward application of the open map theorem, if $f:X\rightarrow Y$ is bijective, it is open (by the open mapping) so its inverse is continuous. To see this, consider $U\subset X$, open, $(f^{-1})^{-1}(U)=f(U)$ is open since $f$ is open. This implies that $f^{-1}$ is continue.

https://en.wikipedia.org/wiki/Open_mapping_theorem_%28functional_analysis%29

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