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. More precisely, (Rudin 1973, Theorem 2.11) harv error: no target: CITEREFRudin1973 (help):

Open mapping theorem for Banach spaces

Open mapping theorem for Banach spaces — If X and Y are Banach spaces and A : X → 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 → 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 ${\displaystyle U=B_{1}^{X}(0),V=B_{1}^{Y}(0)}$. 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 c ∈ Y and r > 0 such that

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

Let v ∈ V, 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 y ∈ Y and all 𝜀 > 0, there exists some x ∈ X 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 V ⊆ A(2LU).

Let y ∈ V. By (1), there is some x₁ with ||x₁|| < L and ||y − Ax₁|| < 1/2. Define a sequence (x_n) 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 x_n+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 x_n+1. Let

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

From the first inequality in (2), {s_n} is a Cauchy sequence, and since X is complete, s_n converges to some x ∈ X. By (2), the sequence As_n 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 V ⊆ A(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 0 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 → Y 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. ${\displaystyle {\overline {T\left(B_{X}\right)}}\subseteq \delta B_{Y}}$;
3. T(B_X) ⊆ δ 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 : X → Y is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A ⁻¹ : Y → X is continuous as well (this is called the bounded inverse theorem).(Rudin 1973, Corollary 2.12) harv error: no target: CITEREFRudin1973 (help)
• If A : X → Y is a linear operator between the Banach spaces X and Y, and if for every sequence (x_n) in X with x_n → 0 and Ax_n → y it follows that y = 0, then A is continuous (the closed graph theorem).(Rudin 1973, Theorem 2.15) harv error: no target: CITEREFRudin1973 (help)

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, Theorem 2.11) harv error: no target: CITEREFRudin (help)) — Let X be a F-space and Y a topological vector space. If A : X → 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

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

where X / N is the quotient space (also an F-space) of X by the closed subspace N. The quotient mapping X → X / N is open, and the mapping α is an isomorphism of topological vector spaces (Dieudonné, 12.16.8) harv error: no target: CITEREFDieudonné (help).

Open mapping theorem ([2]) — If A : X → 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:

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 ([2]) — Let A : X → 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 → Y is a surjective open map and Y is a complete pseudometrizable TVS.

The open mapping theorem can also be stated as

Theorem[3] — 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 → Y 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[4] — If A : X → 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 → Y is a homeomorphism (and thus an isomorphism of TVSs).

Webbed spaces

Main article: Webbed space

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

See also

• Almost open linear map
• Bounded inverse theorem
• Closed graph
• Closed graph theorem
• Open mapping theorem (complex analysis)
• Surjection of Fréchet spaces
• Webbed space

References

1. Rudin 1991, p. 100. sfn error: no target: CITEREFRudin1991 (help)
2. Narici 2011, p. 468. sfn error: no target: CITEREFNarici2011 (help)
3. Trèves (1967) harvtxt error: no target: CITEREFTrèves1967 (help), p. 170
4. Narici 2011, p. 469. sfn error: no target: CITEREFNarici2011 (help)

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