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Concept
Open mapping theorem (functional analysis)
Summary
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
This proof uses the , and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if and are taken to be Fréchet spaces.
Suppose is a surjective continuous linear operator. In order to prove that is an open map, it is sufficient to show that maps the open unit ball in to a neighborhood of the origin of
Let
Then
Since is surjective:
But is Banach so by
That is, we have and such that
Let then
By continuity of addition and linearity, the difference satisfies
and by linearity again,
where we have set
It follows that for all and all there exists some such that
Our next goal is to show that
Let
By (1), there is some with and
Define a sequence inductively as follows.
Assume:
Then by (1) we can pick so that:
so (2) is satisfied for Let
From the first inequality in (2), is a Cauchy sequence, and since is complete, converges to some
By (2), the sequence tends to and so by continuity of
Also,
This shows that belongs to so as claimed.
Thus the image of the unit ball in contains the open ball of
Hence, is a neighborhood of the origin in and this concludes the proof.
The open mapping theorem has several important consequences:
If is a bijective continuous linear operator between the Banach spaces and then the inverse operator is continuous as well (this is called the bounded inverse theorem).
If is a linear operator between the Banach spaces and and if for every sequence in with and it follows that then is continuous (the closed graph theorem).
Local convexity of or is not essential to the proof, but completeness is: the theorem remains true in the case when and are F-spaces.
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