Banach open mapping theorem
This file contains the Banach open mapping theorem, i.e., the fact that a bijective bounded linear map between Banach spaces has a bounded inverse.
First step of the proof of the Banach open mapping theorem (using completeness of
by Baire's theorem, there exists a ball in
E whose image closure has nonempty interior.
Rescaling everything, it follows that any
y ∈ F is arbitrarily well approached by
images of elements of norm at most
C * ∥y∥.
For further use, we will only need such an element whose image
is within distance
y, to apply an iterative process.
The Banach open mapping theorem: if a bounded linear map between Banach spaces is onto, then any point has a preimage with controlled norm.
The Banach open mapping theorem: a surjective bounded linear map between Banach spaces is open.
If a bounded linear map is a bijection, then its inverse is also a bounded linear map.
Associating to a linear equivalence between Banach spaces a continuous linear equivalence when the direct map is continuous, thanks to the Banach open mapping theorem that ensures that the inverse map is also continuous.
Convert a bijective continuous linear map
f : E →L[𝕜] F between two Banach spaces
to a continuous linear equivalence.