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

• May 24th 2010, 03:53 AM
ramdayal9
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?
• May 24th 2010, 04:08 PM
Drexel28
Quote:

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 $\displaystyle T(B_1(0))$ contains an open ball around zero?
• May 25th 2010, 01:15 AM
ramdayal9
Quote:

Originally Posted by Drexel28
I don't understand what the question is asking. Are you saying that $\displaystyle T(B_1(0))$ contains an open ball around zero?

yes (Nod)