Proving an equivalent statement of open mapping theorem

Question

I am looking for guidance on how to prove this particular statement of the open mapping theorem. I know of proofs for equivalent statements.

Prove: if $T: X \rightarrow Y$ is a one-to-one, onto bounded linear map between Banach space $X,Y$, then $T^{-1}: Y \rightarrow X$ is bounded.

Answer 1

For a linear map continuity is equivalent to boundedness. Open mapping theorem immediately tells you that $T^{-1}$ is continous since the inverse image of an open set $U$ under it is nothing but $T(U)$.