Open mapping theorem (functional analysis)

Last updated

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 : XY 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 : XY 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 ${\displaystyle U=B_{1}^{X}(0),V=B_{1}^{Y}(0).}$ Then

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

Since A is surjective:

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

But Y is Banach so by Baire's category theorem

${\displaystyle \exists k\in \mathbb {N}$ :\qquad \left({\overline {A(kU)}}\right)^{\circ }\neq \varnothing }.

That is, we have cY and r > 0 such that

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

Let vV, then

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

By continuity of addition and linearity, the difference rv satisfies

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

and by linearity again,

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

where we have set L=2k/r. It follows that for all yY and all 𝜀 > 0, there exists some xX such that

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

Our next goal is to show that VA(2LU).

Let yV. By (1), there is some x1 with ||x1|| < L and ||yAx1|| < 1/2. Define a sequence (xn) inductively as follows. Assume:

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

Then by (1) we can pick xn+1 so that:

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

so (2) is satisfied for xn+1. Let

${\displaystyle s_{n}=x_{1}+x_{2}+\cdots +x_{n}}$.

From the first inequality in (2), {sn} is a Cauchy sequence, and since X is complete, sn converges to some xX. By (2), the sequence Asn tends to y, and so Ax = y by continuity of A. Also,

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

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

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

1. ${\displaystyle \left\|T^{*}y^{*}\right\|\geq \delta \left\|y^{*}\right\|}$ for all ${\displaystyle y^{*}\in Y^{*}}$;
2. ${\displaystyle {\overline {T\left(B_{X}\right)}}\supseteq \delta B_{Y}}$;
3. ${\displaystyle {T\left(B_{X}\right)}\supseteq \delta B_{Y}}$;
4. 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 : XY is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A−1 : YX is continuous as well (this is called the bounded inverse theorem). [2]
• If A : XY is a linear operator between the Banach spaces X and Y, and if for every sequence (xn) in X with xn → 0 and Axny 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 : XY 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

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

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

Open mapping theorem ( [5] )  If A : XY 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:

1. Y is a Baire space, or
2. 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 : XY 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 : XY 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 : XY is a topological vector space (TVS) homomorphism if the induced map ${\displaystyle {\hat {u}}:X/\ker(u)\to Y}$ is a TVS-isomorphism onto its image.

Consequences

Theorem [7]   If A : XY is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then A : XY 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.

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → ℝ so that

1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
2. Addition in V is continuous with respect to d.
3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
4. The metric space (V, d) is complete.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In particular, they give conditions when functions with closed graphs are necessarily continuous. In mathematics, there are several results known as the "closed graph theorem".

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related areas of mathematics, a barrelled space is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.

In mathematics, the bounded inverse theorem is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed graph theorem.

In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : XY between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.

In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

In functional analysis and related areas of mathematics, a metrizable topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In functional analysis and related areas of mathematics, an almost open map between topological spacess is a map that satisfies a condition similar to, but weaker than, the condition of being an open map.

References

1. Rudin 1991, p. 100.
2. Rudin 1973, Corollary 2.12.
3. Rudin 1973, Theorem 2.15.
4. Dieudonné 1970, 12.16.8.
5. Narici & Beckenstein 2011, p. 468.
6. Trèves 2006, p. 170
7. Narici & Beckenstein 2011, p. 469.