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Math Help - Open map theorem

  1. #1
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    Open map theorem

    Hello!
    I wanted to ask something about functional analysis but i looks that this topic doesn't exist. I hope it is ok to publish it here.
    Well, I wanted to proove that:
    Let X and Y be two Banach spaces and "f" a map between the, such tahta it is linear, continuous and surjerctive. Then:
    If K is a compact subset of Y then there exists L which is a compact subset of X such that f(L)=K.

    Well, I know I have to apply that because of the open map theorem T is an open map but I am not sure about what to do.
    Any ideas???
    Thank you very much!!
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  2. #2
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    Re: Open map theorem

    Quote Originally Posted by Lolyta View Post
    Hello!
    I wanted to ask something about functional analysis but i looks that this topic doesn't exist. I hope it is ok to publish it here.
    Well, I wanted to proove that:
    Let X and Y be two Banach spaces and "f" a map between the, such tahta it is linear, continuous and surjerctive. Then:
    If K is a compact subset of Y then there exists L which is a compact subset of X such that f(L)=K.

    Well, I know I have to apply that because of the open map theorem T is an open map but I am not sure about what to do.
    Any ideas???
    Thank you very much!!
    perhaps consider the mapping of the complements of L and K
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