# 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.${\ 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 .