Open mapping theorem (functional analysis)

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

Classical (Banach space) form

Open mapping theorem for Banach spaces (Rudin 1973, Theorem 2.11) — If $X$ and $Y$ are Banach spaces and $A : X \to Y$ is a surjective continuous linear operator, then $A$ is an open map (i.e. if $U$ is an open set in $X$ , then $A (U)$ is open in $Y$ ).

One proof uses Baire's category theorem, and completeness of both $X$ and $Y$ 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 $X$ and $Y$ are taken to be Fréchet spaces.

Proof

Suppose $A : X \to Y$ is a surjective continuous linear operator. In order to prove that $A$ is an open map, it is sufficient to show that $A$ maps the open unit ball in $X$ to a neighborhood of the origin of $Y$ .

Let $U=B_{1}^{X}(0),V=B_{1}^{Y}(0).$ Then

X=\bigcup _{k\in \mathbb {N} }kU.

Since $A$ is surjective:

Y=A(X)=A\left(\bigcup _{k\in \mathbb {N} }kU\right)=\bigcup _{k\in \mathbb {N} }A(kU).

But $Y$ is Banach so by Baire's category theorem

\exists k\in \mathbb {N

.

That is, we have $c \in Y$ and $r > 0$ such that

B_{r}(c)\subseteq \left({\overline {A(kU)}}\right)^{\circ }\subseteq {\overline {A(kU)}}

.

Let $v \in V$ , then

c,c+rv\in B_{r}(c)\subseteq {\overline {A(kU)}}.

By continuity of addition and linearity, the difference $rv$ satisfies

rv\in {\overline {A(kU)}}+{\overline {A(kU)}}\subseteq {\overline {A(kU)+A(kU)}}\subseteq {\overline {A(2kU)}},

and by linearity again,

V\subseteq {\overline {A\left(LU\right)}}

where we have set $L =2 k / r$ . It follows that for all $y \in Y$ and all $𝜀 > 0$ , there exists some $x \in X$ such that

\qquad \|x\|_{X}\leq L\|y\|_{Y}\quad {\text{and}}\quad \|y-Ax\|_{Y}<\epsilon .\qquad (1)

Our next goal is to show that $V \subseteq A (2 LU)$ .

Let $y \in V$ . By (1), there is some $x 1$ with $|| x 1 || < L$ and $|| y - Ax 1 || < 1/2$ . Define a sequence $(x n)$ inductively as follows. Assume:

\|x_{n}\|<{\frac {L}{2^{n-1}}}\quad {\text{and}}\quad \left\|y-A\left(x_{1}+x_{2}+\cdots +x_{n}\right)\right\|<{\frac {1}{2^{n}}}.\qquad (2)

Then by (1) we can pick $x n +1$ so that:

\|x_{n+1}\|<{\frac {L}{2^{n}}}\quad {\text{and}}\quad \left\|y-A\left(x_{1}+x_{2}+\cdots +x_{n}\right)-A\left(x_{n+1}\right)\right\|<{\frac {1}{2^{n+1}}},

so (2) is satisfied for $x n +1$ . Let

s_{n}=x_{1}+x_{2}+\cdots +x_{n}

.

From the first inequality in (2), {s_n} is a Cauchy sequence, and since $X$ is complete, $s n$ converges to some $x \in X$ . By (2), the sequence $As n$ tends to $y$ , and so $Ax = y$ by continuity of $A$ . Also,

\|x\|=\lim _{n\to \infty }\|s_{n}\|\leq \sum _{n=1}^{\infty }\|x_{n}\|<2L.

This shows that $y$ belongs to $A (2 LU)$ , so $V \subseteq A (2 LU)$ as claimed. Thus the image A(U) of the unit ball in $X$ contains the open ball $V /2 L$ of $Y$ . Hence, $A (U)$ is a neighborhood of the origin in $Y$ , and this concludes the proof.

Related results

Theorem[1] — Let $X$ and $Y$ be Banach spaces, let $B X$ and $B Y$ denote their open unit balls, and let $T : X \to Y$ be a bounded linear operator. If $δ > 0$ then among the following four statements we have $(1)\implies (2)\implies (3)\implies (4)$ (with the same $δ$ )

$\left\|T^{*}y^{*}\right\|\geq \delta \left\|y^{*}\right\|$ for all $y^{*}\in Y^{*}$ ;
${\overline {T\left(B_{X}\right)}}\subseteq \delta B_{Y}$ ;
$T (B X) \subseteq δ B Y$ ;
$Im T = Y$ (i.e. $T$ is surjective).

Furthermore, if $T$ is surjective then (1) holds for some $δ > 0$

Consequences

The open mapping theorem has several important consequences:

If $A : X \to Y$ is a bijective continuous linear operator between the Banach spaces $X$ and $Y$ , then the inverse operator $A -1 : Y \to X$ is continuous as well (this is called the bounded inverse theorem).[2]
If $A : X \to Y$ is a linear operator between the Banach spaces $X$ and $Y$ , and if for every sequence $(x n)$ in $X$ with $x n \to 0$ and $Ax n \to y$ it follows that y = 0, then $A$ is continuous (the closed graph theorem).[3]

Generalizations

Local convexity of $X$ or $Y$ is not essential to the proof, but completeness is: the theorem remains true in the case when $X$ and $Y$ are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:

Theorem ((Rudin 1991, Theorem 2.11)) — Let $X$ be a F-space and $Y$ a topological vector space. If $A : X \to Y$ is a continuous linear operator, then either $A (X)$ is a meager set in Y, or $A (X) = Y$ . In the latter case, $A$ is an open mapping and $Y$ is also an F-space.

Furthermore, in this latter case if $N$ is the kernel of $A$ , then there is a canonical factorization of $A$ in the form

X\to X/N{\overset {\alpha }{\to }}Y

where $X / N$ is the quotient space (also an F-space) of $X$ by the closed subspace $N$ . The quotient mapping $X \to X / N$ is open, and the mapping α is an isomorphism of topological vector spaces.[4]

Open mapping theorem ([5]) — If $A : X \to Y$ is a surjective closed linear operator from an complete pseudometrizable TVS $X$ into a topological vector space $Y$ and if at least one of the following conditions is satisfied:

$Y$ is a Baire space, or
$X$ is locally convex and $Y$ is a barrelled space,

either $A (X)$ is a meager set in Y, or $A (X) = Y$ . then $A$ is an open mapping.

Open mapping theorem for continuous maps ([5]) — Let $A : X \to Y$ be a continuous linear operator from an complete pseudometrizable TVS $X$ into a Hausdorff topological vector space $Y$ . If $Im A$ is nonmeager in $Y$ then $A : X \to Y$ is a surjective open map and $Y$ is a complete pseudometrizable TVS.

The open mapping theorem can also be stated as

Theorem[6] — Let $X$ and $Y$ be two F-spaces. Then every continuous linear map of $X$ onto $Y$ is a TVS homomorphism, where a linear map $u : X \to Y$ is a topological vector space (TVS) homomorphism if the induced map ${\hat {u}}:X/\ker(u)\to Y$ is a TVS-isomorphism onto its image.

Consequences

Theorem[7] — If $A : X \to Y$ is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then $A : X \to Y$ is a homeomorphism (and thus an isomorphism of TVSs).

Webbed spaces

Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.

References

Rudin 1991, p. 100.
Rudin 1973, Corollary 2.12.
Rudin 1973, Theorem 2.15.
Dieudonné 1970, 12.16.8.
Narici & Beckenstein 2011, p. 468.
Trèves 2006, p. 170
Narici & Beckenstein 2011, p. 469.

Bibliography

Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
Dieudonné, Jean (1970), Treatise on Analysis, Volume II, Academic Press
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259.
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

This article incorporates material from Proof of open mapping theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[FOOTNOTERudin1991100-1] Rudin 1991, p. 100.

[FOOTNOTERudin1973Corollary_2.12-2] Rudin 1973, Corollary 2.12.

[FOOTNOTERudin1973Theorem_2.15-3] Rudin 1973, Theorem 2.15.

[FOOTNOTEDieudonné197012.16.8-4] Dieudonné 1970, 12.16.8.

[FOOTNOTENariciBeckenstein2011468-5] Narici & Beckenstein 2011, p. 468.

[6] Trèves 2006, p. 170

[FOOTNOTENariciBeckenstein2011469-7] Narici & Beckenstein 2011, p. 469.

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property

Open mapping theorem (functional analysis)

Classical (Banach space) form

Related results

Consequences

Generalizations

Consequences

Webbed spaces

See also

References

Bibliography