In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem[1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
Classical (Banach space) form
editOpen mapping theorem for Banach spaces (Rudin 1973, Theorem 2.11) — If and
are Banach spaces and
is a surjective continuous linear operator, then
is an open map (that is, if
is an open set in
then
is open in
).
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 vector space, but is true if
and
are taken to be Fréchet spaces.
Proof |
---|
Suppose Let Since But That is, we have Let By continuity of addition and linearity, the difference where we have set Our next goal is to show that Let Then by (1) we can pick From the first inequality in (2), This shows that |
Related results
editTheorem[2] — Let and
be Banach spaces, let
and
denote their open unit balls, and let
be a bounded linear operator.If
then among the following four statements we have
(with the same
)
for all
;
;
;
(that is,
is surjective).
Furthermore, if is surjective then (1) holds for some
Consequences
editThe 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).[3]
- 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).[4]
Generalizations
editLocal 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:
Open mapping theorem for continuous maps[5][6] — Let be a continuous linear operator from a complete pseudometrizable TVS
onto a Hausdorff TVS
If
is nonmeager in
then
is a (surjective) open map and
is a complete pseudometrizable TVS.Moreover, if
is assumed to be hausdorff (i.e. a F-space), then
is also an F-space.
Furthermore, in this latter case if is the kernel of
then there is a canonical factorization of
in the form
An important special case of this theorem can also be stated as
Theorem[8] — Let and
be two F-spaces. Then every continuous linear map of
onto
is a TVS homomorphism,where a linear map
is a topological vector space (TVS) homomorphism if the induced map
is a TVS-isomorphism onto its image.
On the other hand, a more general formulation, which implies the first, can be given:
Open mapping theorem[6] — Let be a surjective linear map from a complete pseudometrizable TVS
onto a TVS
and suppose that at least one of the following two conditions is satisfied:
is a Baire space, or
is locally convex and
is a barrelled space,
If is a closed linear operator then
is an open mapping.If
is a continuous linear operator and
is Hausdorff then
is (a closed linear operator and thus also) an open mapping.
Nearly/Almost open linear maps
A linear map between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood
of the origin in the domain, the closure of its image
is a neighborhood of the origin in
[9] 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
[9] but for surjective maps these definitions are equivalent.A bijective linear map is nearly open if and only if its inverse is continuous.[9]Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.[10] The same is true of every surjective linear map from a TVS onto a Baire TVS.[10]
Open mapping theorem[11] — If a closed surjective linear map from a complete pseudometrizable TVS onto a Hausdorff TVS is nearly open then it is open.
Consequences
editTheorem[12] — If is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then
is a homeomorphism (and thus an isomorphism of TVSs).
Webbed spaces
editWebbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.
See also
edit- Almost open linear map – Map that satisfies a condition similar to that of being an open map.
- Bounded inverse theorem
- Closed graph – Graph of a map closed in the product space
- Closed graph theorem – Theorem relating continuity to graphs
- Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
- Open mapping theorem (complex analysis) – Theorem that holomorphic functions on complex domains are open maps
- Surjection of Fréchet spaces – Characterization of surjectivity
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
- Webbed space – Space where open mapping and closed graph theorems hold
References
edit- ^ Trèves 2006, p. 166.
- ^ Rudin 1991, p. 100.
- ^ Rudin 1973, Corollary 2.12.
- ^ Rudin 1973, Theorem 2.15.
- ^ Rudin 1991, Theorem 2.11.
- ^ a b Narici & Beckenstein 2011, p. 468.
- ^ Dieudonné 1970, 12.16.8.
- ^ Trèves 2006, p. 170
- ^ a b c Narici & Beckenstein 2011, pp. 466.
- ^ a b Narici & Beckenstein 2011, pp. 467.
- ^ Narici & Beckenstein 2011, pp. 466−468.
- ^ Narici & Beckenstein 2011, p. 469.
Bibliography
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