I was trying the following question :
Let $V_1$ and $V_2$ be two vector spaces such that there exists linear transformations $T_1 : V_1 \to V_2$ and $T_2 : V_2 \to V_1$ both onto. Then does it imply that $V_1$ and $V_2$ are isomorphic as vector spaces?
When $V_1$ and $V_2$ are finite dimensional then it's very easy to prove that the answer is affirmative.
But I am confused about the case where both are infinite dimensional. I really don't have any idea in this case.
Thanks in advance for help!