linear map open iff image of unit ball contains ball around 0

Mar 2009
17
1
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?
 

Drexel28

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Nov 2009
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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 \(\displaystyle T(B_1(0))\) contains an open ball around zero?