Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that is bounded.

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- June 5th 2011, 06:39 PMgeezeelaLinear operator
Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that is bounded.

- June 5th 2011, 06:45 PMbryangoodrich
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- June 5th 2011, 07:19 PMDrexel28
Since, as the user above me has pointed out, you need to show effort I will tell you a more general fact that you may have seen before from topology (something you probalby have taken) which answers this question (with a little work) nicely "Let be a continous bijection where is compact and Hausdorff. Then, is a homeomorphism."

- June 5th 2011, 07:52 PMgeezeela
if is homeomorphism the X and Y and homeomorphic.F^-1 is continuous but how can we prove that it is bounded?

- June 5th 2011, 08:02 PMDrexel28
- June 5th 2011, 09:01 PMgeezeela
Let be a continous bijection where is compact and Hausdorff. Then, is a homeomorphism since is is homeomorphism then X and Y and homeomorphic.F^-1 is continuous which conculdes that F^-1 is bounded. Is that all i have to say??

- June 5th 2011, 09:02 PMDrexel28
- June 5th 2011, 09:16 PMgeezeela
- June 5th 2011, 09:17 PMDrexel28
I would suggest looking at this blog post of mine and the next two (press 'next' at the bottom of the page to go the next post).

- June 5th 2011, 09:34 PMgeezeela
wow...it helped me a lot...thank you ver much my friend...:))

- June 5th 2011, 09:35 PMDrexel28
- June 5th 2011, 10:41 PMJose27