Theorem of open mapping (functional analysis)

This article deals with the theorem from the mathematical branch of functional analysis. For more sentences of this name see the sentence of the open figure .

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. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle Y}$
${\displaystyle T}$ ${\displaystyle T}$

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 . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

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. ${\displaystyle T}$

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. ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Tx=b}$ ${\displaystyle b}$ ${\displaystyle Y}$ ${\displaystyle x}$ ${\displaystyle b}$

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

1. ↑ 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 .