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Math Help - Linear operator

  1. #1
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    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.
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    Forum Rule 11

    11. Show some effort. If you want help with a question it is expected that you will show some effort. Effort might include showing your working, taking the time to learn how to typeset equations using latex (there is an entire subforum devoted to this), formatting your question so that it is more easily understood, using effective post titles and posting in the appropriate subforum, making a genuine attempt to understand the help that is given before asking for more help and learning from previous questions asked. Moderators reserve the right to Close threads in cases where the member is not making a genuine effort (particularly if the member is spamming the forums with multiple questions of exactly the same type). It should also be remembered that all contributors to MHF are unpaid volunteers and are under no obligation to answer a question.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    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."
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    if f is homeomorphism the X and Y and homeomorphic.F^-1 is continuous but how can we prove that it is bounded?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    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?
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  6. #6
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    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??
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    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.
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    Quote Originally Posted by Drexel28 View Post
    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
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    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).
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  10. #10
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    wow...it helped me a lot...thank you ver much my friend...)
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  11. #11
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    wow...it helped me a lot...thank you ver much my friend...)
    Good, I'm glad it did.
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  12. #12
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    Quote Originally Posted by geezeela View Post
    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)
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