1. ## Linear operator

Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that $\displaystyle {T}^{-1}$ is bounded.

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3. Originally Posted by geezeela
Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that $\displaystyle {T}^{-1}$ is bounded.
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 $\displaystyle f:X\to Y$ be a continous bijection where $\displaystyle X$ is compact and $\displaystyle Y$ Hausdorff. Then, $\displaystyle f$ is a homeomorphism."

4. if $\displaystyle f$ is homeomorphism the X and Y and homeomorphic.F^-1 is continuous but how can we prove that it is bounded?

5. Originally Posted by geezeela
if $\displaystyle f$ is homeomorphism the X and Y and homeomorphic.F^-1 is continuous but how can we prove that it is bounded?
Isn't boundedness equivalent to continuity for linear maps?

6. Let $\displaystyle f:X\to Y$ be a continous bijection where $\displaystyle X$ is compact and $\displaystyle Y$ Hausdorff. Then, $\displaystyle f$ is a homeomorphism since $\displaystyle f$ 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??

7. Originally Posted by geezeela
Let $\displaystyle f:X\to Y$ be a continous bijection where $\displaystyle X$ is compact and $\displaystyle Y$ Hausdorff. Then, $\displaystyle f$ is a homeomorphism since $\displaystyle f$ 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??
Right--assuming you know the topological fact and the fact that continuity is equivalent to boundedness for linear maps between normed spaces.

8. Originally Posted by Drexel28
Right--assuming you know the topological fact and the fact that continuity is equivalent to boundedness for linear maps between normed spaces.
i havent done any topology yet in my life. That why i am facing problem with this questions. Please can you explain to me how the continuity is equivalent to boundedness for linear maps between normed spaces. Thank you very much

9. Originally Posted by geezeela
i havent done any topology yet in my life. That why i am facing problem with this questions. Please can you explain to me how the continuity is equivalent to boundedness for linear maps between normed spaces. Thank you very much
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).

10. wow...it helped me a lot...thank you ver much my friend...)

11. Originally Posted by geezeela
wow...it helped me a lot...thank you ver much my friend...)
Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that $\displaystyle {T}^{-1}$ is bounded.