bijective bounded linear operators have bounded linear inverses

If $B_1,B_2$ are Banach spaces and $T:B_1\to B_2$ is a bijective bounded linear operator, then $T^{-1}\in \mathcal{B}(B_2,B_1)$ is also.

$T^{-1}$ is continuous $⟺$ for all $U\subset B_1$ open, $(T^{-1}{)}^{-1}(U)=T(U)$ is open. And this is true by the open mapping theorem