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Math Help - linear map open iff image of unit ball contains ball around 0

  1. #1
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    linear map open iff image of unit ball contains ball around 0

    Let X, Y be normed spaces. Prove that a map T: X ---> Y is open if and only if T(B(0,1)) contains a ball around 0 \in Y

    I think I can prove --> direction.
    But I am struggling to show the map is open. I take an open set U in X. I have to show T(U) is open. How would I proceed from here?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ramdayal9 View Post
    Let X, Y be normed spaces. Prove that a map T: X ---> Y is open if and only if T(B(0,1)) contains a ball around 0 \in Y

    I think I can prove --> direction.
    But I am struggling to show the map is open. I take an open set U in X. I have to show T(U) is open. How would I proceed from here?
    I don't understand what the question is asking. Are you saying that T(B_1(0)) contains an open ball around zero?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    I don't understand what the question is asking. Are you saying that T(B_1(0)) contains an open ball around zero?
    yes
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