Open mapping theorem (functional analysis)

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 $$X=\bigcup _{k\in \mathbb {N} }kU.$$ 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 $v\in V,$ 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 $\epsilon >0,$ 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 $V\subseteq A(2LU).$ Let $y\in V.$ By (1), there is some $x_{1}$ with $\left\|x_{1}\right\|<L$ and $\left\|y-Ax_{1}\right\|<1/2.$ Define a sequence $\left(x_{n}\right)$ 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), $\left(s_{n}\right)$is a Cauchy sequence, and since $X$ is complete, $s_{n}$ converges to some $x\in X.$ By (2), the sequence $As_{n}$ 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 $V\subseteq A(2LU)$ 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 $(1)\implies (2)\implies (3)\implies (4)$ (with the same $\delta$)

1. $\left\|T^{*}y^{*}\right\|\geq \delta \left\|y^{*}\right\|$ 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. $\operatorname {Im} T=Y$ (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 $Ax_{n}\to y$ 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 $\operatorname {Im} A$ 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 $$X\to X/N{\overset {\alpha }{\to }}Y$$ 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 $\alpha$ 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 $\operatorname {cl} A(U)$ 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 – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Bounded inverse theorem – Condition for a linear operator to be openPages displaying short descriptions of redirect targets
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• 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) – Theorem that holomorphic functions on complex domains are open mapsPages displaying wikidata descriptions as a fallback
• 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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