Thread: 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?

Originally Posted by ramdayal9

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 contains an open ball around zero?

Originally Posted by Drexel28

I don't understand what the question is asking. Are you saying that contains an open ball around zero?

yes