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):

Open Mapping Theorem. 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).

The proof uses the Baire 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.

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)
• 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)

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 $U=B_1^X(0), V=B_1^Y(0)$. 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

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

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 ∈ 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=2k/r. It follows that

$\forall y \in Y, \forall \varepsilon > 0, \exists x \in X: \qquad \|x\|_X \le L \|y\|_Y \quad \text{and} \quad \|y - Ax\|_X < \varepsilon. \qquad (1)$

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:

$\|x_n\| < \frac{L}{2^{n-1}} \quad \text{and} \quad \left\|y - A(x_1+x_2+ \cdots +x_n) \right \| < \frac{1}{2^{n}}. \qquad (2)$

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

$\|x_{n+1}\| < \frac{L}{2^n} \quad \text{and} \quad \left \|y - A(x_1+x_2+ \cdots +x_n) - A(x_{n+1}) \right \| < \frac{1}{2^{n+1}},$

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 ∈ 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(2LU), so V ⊆ A(2LU) as claimed. Thus the image A(U) of the unit ball in X contains the open ball V/2LU of Y. Hence, A(U) is a neighborhood of 0 in Y, and this concludes the proof.

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 (Rudin, Theorem 2.11):

• 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

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

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).

The open mapping theorem can also be stated as

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 $\hat{u} : X/\ker(u) \to Y$ is a TVS-isomorphism onto its image.

See also

• Open mapping theorem (complex analysis)

References

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