Open mapping theorem (functional analysis)

Short description: Condition for a linear operator to be open

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or 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 (that is, if $U$ is an open set in $X,$ then $A(U)$ is open in $Y$).

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

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

But $Y$ is Banach so by Baire's category theorem $$\mathrm{\exists }k\in \mathbb{\mathbb{N}}:{\left(\overline{A(kU)}\right)}^{\circ }\ne \varnothing .$$

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 $v\in V,$ 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, $$V\subseteq \overline{A(LU)}$$

where we have set $L=2k/r.$ It follows that for all $y\in Y$ and all $ϵ>0,$ there exists some $x\in X$ such that $$\Vert x{\Vert }_X\le L\Vert y{\Vert }_Y\text{and}\Vert y-Ax{\Vert }_Y<ϵ.(1)$$

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

Let $y\in V.$ By (1), there is some $x_1$ with $\Vert x_1\Vert <L$ and $\Vert y-A{x}_1\Vert <1/2.$ Define a sequence $\left(x_n\right)$ inductively as follows. Assume: $$\Vert x_n\Vert <\frac{L}{2^{n-1}}\text{and}\Vert y-A(x_1+x_2+\cdots +x_n)\Vert <\frac{1}{2^n}.(2)$$

Then by (1) we can pick $x_{n+1}$ so that: $$\Vert x_{n+1}\Vert <\frac{L}{2^n}\text{and}\Vert y-A(x_1+x_2+\cdots +x_n)-A\left(x_{n+1}\right)\Vert <\frac{1}{2^{n+1}},$$ so (2) is satisfied for $x_{n+1}.$ Let $$s_n=x_1+x_2+\cdots +x_n.$$

From the first inequality in (2), $\left(s_n\right)$is a Cauchy sequence, and since $X$ is complete, $s_n$ converges to some $x\in X.$ By (2), the sequence $A{s}_n$ tends to $y$ and so $Ax=y$ by continuity of $A.$ Also, $$\Vert x\Vert =\underset{n\to \mathrm{\infty }}{\lim }\Vert s_n\Vert \le \sum _{n=1}^{\mathrm{\infty }}\Vert x_n\Vert <2L.$$

This shows that $y$ belongs to $A(2LU),$ so $V\subseteq A(2LU)$ 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 the origin in $Y,$ and this concludes the proof.

Related results

Theorem — 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 $\delta >0$ then among the following four statements we have $(1)⟹(2)⟹(3)⟹(4)$ (with the same $\delta$)

1. $\Vert T^{*}{y}^{*}\Vert \ge \delta \Vert y^{*}\Vert$ for all $y^{*}\in Y^{*}$;
2. $\overline{T\left(B_X\right)}\supseteq \delta B_Y$;
3. $T\left(B_X\right)\supseteq \delta B_Y$;
4. $\mathrm{Im}T=Y$ (that is, $T$ is surjective).

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

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).
• If $A:X\to Y$ is a linear operator between the Banach spaces $X$ and $Y,$ and if for every sequence $\left(x_n\right)$ in $X$ with $x_n\to 0$ and $A{x}_n\to y$ it follows that $y=0,$ then $A$ is continuous (the closed graph theorem).

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:

Open mapping theorem for continuous maps — Let $A:X\to Y$ be a continuous linear operator from a complete pseudometrizable TVS $X$ onto a Hausdorff TVS $Y.$ If $\mathrm{Im}A$ is nonmeager in $Y$ then $A:X\to Y$ is a (surjective) open map and $Y$ is a complete pseudometrizable TVS. Moreover, if $X$ is assumed to be hausdorff (i.e. a F-space), then $Y$ 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 $$X\to X/N\stackrel{\alpha }{\to }Y$$ 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 $\alpha$ is an isomorphism of topological vector spaces.

An important special case of this theorem can also be stated as

Theorem — 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.

On the other hand, a more general formulation, which implies the first, can be given:

Open mapping theorem — Let $A:X\to Y$ be a surjective linear map from a complete pseudometrizable TVS $X$ onto a TVS $Y$ and suppose that at least one of the following two conditions is satisfied:

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

If $A$ is a closed linear operator then $A$ is an open mapping. If $A$ is a continuous linear operator and $Y$ is Hausdorff then $A$ is (a closed linear operator and thus also) an open mapping.

Nearly/Almost open linear maps

A linear map $A:X\to Y$ between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood $U$ of the origin in the domain, the closure of its image $\mathrm{cl}A(U)$ is a neighborhood of the origin in $Y.$ Many authors use a different definition of "nearly/almost open map" that requires that the closure of $A(U)$ be a neighborhood of the origin in $A(X)$ rather than in $Y,$ but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous. Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open. The same is true of every surjective linear map from a TVS onto a Baire TVS.

Open mapping theorem — If a closed surjective linear map from a complete pseudometrizable TVS onto a Hausdorff TVS is nearly open then it is open.

Consequences

Theorem — 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

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 – Graph of a map closed in the product space
• Closed graph theorem – Theorem relating continuity to graphs
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Open mapping theorem (complex analysis)
• Surjection of Fréchet spaces – Characterization of surjectivity
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
• Webbed space – Space where open mapping and closed graph theorems hold

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