$T_1 : V_1 \to V_2$ 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?

Question

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?

My attempt:

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!

Answer 1

Yes. In the infinite-dimensional case one can still show that any two bases have the same cardinality, and hence define the dimension to be the cardinality of a basis. Your hypothesis implies that $\dim(V_1)\ge\dim(V_2)$ and $\dim(V_2)\ge\dim(V_1)$, hence $\dim(V_1)=\dim(V_2)$, hence $V_1$ and $V_2$ are isomorphic (a bijection between the bases extends to an isomorphism).