Theorem of open mapping (functional analysis)
The open map theorem , also known as Banach-Schauder's theorem, is a fundamental theorem from functional analysis , a branch of mathematics. The theorem is a consequence of Baire's theorem and was proved in 1929 by Stefan Banach and Juliusz Schauder .
statement
A mapping between topological spaces is called open if the image of any open set is open.
The statement of the sentence is:
- If and are Banach spaces , then for every continuous linear mapping between and : is surjective if and only if is open.
It is easy to see that an open linear mapping must be surjective, since there is no real subspace of open; the content of the proposition lies in the statement that every surjective continuous linear mapping is open. The proof requires the completeness of , as well as that of .
Theorem of the continuous inverse
Immediately from the definition of continuity it follows as a corollary :
- If there is a continuous linear bijection between two Banach spaces, then the inverse mapping is continuous.
This statement is known as the theorem of the inverse mapping or theorem of the continuous inverse . It can also be formulated like this:
- Let be a continuous linear operator between two Banach spaces and . If the equation is uniquely solvable for each in , then the solution depends continuously on.
generalization
In the theory of locally convex spaces, the theorem about open mapping can be extended to larger space classes, see space with tissue , ultrabornological space or (LF) space .
Individual evidence
- ↑ Jürgen Heine: Topology and Functional Analysis: Basics of Abstract Analysis with Applications. Oldenbourg, Munich / Vienna 2002, ISBN 3-486-24914-2 .
literature
- Gert K. Pedersen: Analysis Now (Graduate Texts in Mathematics) . Springer-Verlag , Berlin (among others) 1988, ISBN 3-540-96788-5 .
- Lutz Führer: General topology with applications . Vieweg-Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .