§ Bounded inverse theorem
- Theorem: Every bijective bounded linear operator has bounded inverse.
- Equivaently: Every bijective continuous linear operator has continuous inverse.
- Proof: quick corollary of open mapping. Let be bijective bounded linear operator.
- Assuming open mapping, we know that maps opens to open sets. Recall that bounded iff continuous. Thus, we can show that is continuous to show that is bounded.
- We need to show that inverse images of open sets under is open. Specifically that is open for open
- Since is open as is open and is an open map, this means that is open, as . Hence done.