I have seen an example of a continuous linear bijection $f:S\to S$, where $S$ was a normed linear space, such that the inverse function $f^{-1}$ was not continuous,as it was unbounded.The norm on $S$ was not complete .
It was then stated that if $T$ is a Banach space and $g:T\to T$ is a continuous linear bijection, then $g^{-1}$ is continuous, so that $g$ is a homeomorphism.It said this followed from the Uniform Boundedness Principle. I have been unable to see how to do this.
How do you prove that $g^{-1}$ is continuous?