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
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
or
continuous linear operator between
Banach spaces is
surjective then it is an
open map.
Classical (Banach space) form
This proof uses the
Baire category theorem, 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
Baire's category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
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.
Related results
Consequences
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 mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
in
with
and
it follows that
then
is continuous (the
closed graph theorem).
Generalizations
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. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:
Furthermore, in this latter case if
is the
kernel of
then there is a canonical factorization of
in the form
where
is the
quotient space
Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular:
*Quotient space (topology), in case of topological spaces
* Quotient space (linear algebra), in case of vector spaces
*Quotient ...
(also an F-space) of
by the
closed subspace
The quotient mapping
is open, and the mapping
is an
isomorphism of
topological vector spaces.
The open mapping theorem can also be stated as
Nearly/Almost open linear maps
A linear map
between two topological vector spaces (TVSs) is called a (or sometimes, an ) if for every neighborhood
of the origin in the domain, the closure of its image
is a neighborhood of the origin in
Many authors use a different definition of "nearly/almost open map" that requires that the closure of
be a neighborhood of the origin in
rather than in
but for surjective maps these definitions are equivalent.
A bijective linear map is nearly open if and only if its inverse is continuous.
Every surjective linear map from
locally convex TVS onto a
barrelled TVS is
nearly open
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fund ...
. The same is true of every surjective linear map from a TVS onto a
Baire TVS.
Consequences
Webbed spaces
Webbed spaces are a class of
topological vector spaces for which the open mapping theorem and the
closed graph theorem hold.
See also
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References
Bibliography
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{{Topological vector spaces
Articles containing proofs
Theorems in functional analysis