Open 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 ).
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
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
Theorem[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
The open mapping theorem has several important consequences:
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
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:
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
completepseudometrizable TVS onto a TVS and suppose that at least one of the following two conditions is satisfied:
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 (complex analysis) – Theorem that holomorphic functions on complex domains are open mapsPages displaying wikidata descriptions as a fallback
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York:
Springer-Verlag.
ISBN978-3-540-08662-8.
OCLC297140003.
Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer.
ISBN978-0-387-90081-0.
OCLC878109401.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
ISBN978-3-642-64988-2.
MR0248498.
OCLC840293704.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
ISBN978-1584888666.
OCLC144216834.