Linear operator

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June 5th 2011, 06:39 PM
geezeela

Linear operator

Let X and Y be normed spaces and X compact. If T : X--->Y is a bijective linear operator. Show that ${T}^{-1}$ is bounded.
June 5th 2011, 06:45 PM
bryangoodrich

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Quote:

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June 5th 2011, 07:19 PM
Drexel28

Quote:

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 ${T}^{-1}$ 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 $f:X\to Y$ be a continous bijection where $X$ is compact and $Y$ Hausdorff. Then, $f$ is a homeomorphism."
June 5th 2011, 07:52 PM
geezeela

if $f$ 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 PM
Drexel28

Quote:

Originally Posted by geezeela

if $f$ 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?
June 5th 2011, 09:01 PM
geezeela

Let $f:X\to Y$ be a continous bijection where $X$ is compact and $Y$ Hausdorff. Then, $f$ is a homeomorphism since $f$ 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??
June 5th 2011, 09:02 PM
Drexel28

Quote:

Originally Posted by geezeela

Let $f:X\to Y$ be a continous bijection where $X$ is compact and $Y$ Hausdorff. Then, $f$ is a homeomorphism since $f$ 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.
June 5th 2011, 09:16 PM
geezeela

Quote:

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
June 5th 2011, 09:17 PM
Drexel28

Quote:

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).
June 5th 2011, 09:34 PM
geezeela

wow...it helped me a lot...thank you ver much my friend...:))
June 5th 2011, 09:35 PM
Drexel28

Quote:

Originally Posted by geezeela

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

Good, I'm glad it did.
June 5th 2011, 10:41 PM
Jose27

Quote:

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 ${T}^{-1}$ is bounded.

It's curious to note that there is no such X. Even if you meant complete you need Y be a Banach space too. (I'm assuming this is where the 'mistake' is since otherwise, as noted, this is more of a topology question)