Why is $GL(B)$ a Banach Lie Group?
This follows from three simple observations:
The subset $\operatorname{GL}(B) \subset B$ is open, so it is a Banach manifold modeled on $B$ itself.
Multiplication is the restriction of a continuous linear map, hence it is analytic.
Inversion is locally given by the Neumann series, hence it is analytic, too.