Let $$ and $U'$ be vector spaces isomorphic and $V$ and $V'$ be vector spaces isomorphic
We have to show that $(,)$ and $(',')$ are also isomorphic. Where, $(,)$ is the space of all linear transformations from U to V.
In hypothesis in finite dimension, as $(,)$ and $(',')$ are the same dimension, then they have to be isomorphic. But, I can conclude the same in infinite dimension? If yes, how can I define the function between $(,)$ and $(',')$ in order to get such isomorphism?
Given linear maps $f : U' \to U$ and $g : V \to V'$ you get a map $F : L(U, V) \to L(U', V')$ by sending the linear map $T : U \to V$ to the linear map $g \circ T \circ f : U' \to V'$. If both $f$ and $g$ are isomorphisms can you show that $F$ is as well?