In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem^[1] (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

This proof uses the Baire category theorem, and completeness of both ${\displaystyle X}$ and ${\displaystyle Y}$ 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are taken to be Fréchet spaces.

Proof

Suppose ${\displaystyle A:X\to Y}$ is a surjective continuous linear operator. In order to prove that ${\displaystyle A}$ is an open map, it is sufficient to show that ${\displaystyle A}$ maps the open unit ball in ${\displaystyle X}$ to a neighborhood of the origin of ${\displaystyle Y.}$ Let ${\displaystyle U=B_{1}^{X}(0),V=B_{1}^{Y}(0).}$ Then $${\displaystyle X=\bigcup _{k\in \mathbb {N} }kU.}$$ Since ${\displaystyle A}$ is surjective: $${\displaystyle Y=A(X)=A\left(\bigcup _{k\in \mathbb {N} }kU\right)=\bigcup _{k\in \mathbb {N} }A(kU).}$$ But ${\displaystyle Y}$ 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 ${\displaystyle c\in Y}$ and ${\displaystyle r>0}$ such that $${\displaystyle B_{r}(c)\subseteq \left({\overline {A(kU)}}\right)^{\circ }\subseteq {\overline {A(kU)}}.}$$ Let ${\displaystyle v\in V,}$ then $${\displaystyle c,c+rv\in B_{r}(c)\subseteq {\overline {A(kU)}}.}$$ By continuity of addition and linearity, the difference ${\displaystyle rv}$ satisfies $${\displaystyle rv\in {\overline {A(kU)}}+{\overline {A(kU)}}\subseteq {\overline {A(kU)+A(kU)}}\subseteq {\overline {A(2kU)}},}$$ and by linearity again, $${\displaystyle V\subseteq {\overline {A(LU)}}}$$ where we have set ${\displaystyle L=2k/r.}$ It follows that for all ${\displaystyle y\in Y}$ and all ${\displaystyle \epsilon >0,}$ there exists some ${\displaystyle x\in X}$ 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 ${\displaystyle V\subseteq A(2LU).}$ Let ${\displaystyle y\in V.}$ By (1), there is some ${\displaystyle x_{1}}$ with ${\displaystyle \left\|x_{1}\right\|<L}$ and ${\displaystyle \left\|y-Ax_{1}\right\|<1/2.}$ Define a sequence ${\displaystyle \left(x_{n}\right)}$ 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 ${\displaystyle x_{n+1}}$ 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 ${\displaystyle x_{n+1}.}$ Let $${\displaystyle s_{n}=x_{1}+x_{2}+\cdots +x_{n}.}$$ From the first inequality in (2), ${\displaystyle \left(s_{n}\right)}$is a Cauchy sequence, and since ${\displaystyle X}$ is complete, ${\displaystyle s_{n}}$ converges to some ${\displaystyle x\in X.}$ By (2), the sequence ${\displaystyle As_{n}}$ tends to ${\displaystyle y}$ and so ${\displaystyle Ax=y}$ by continuity of ${\displaystyle A.}$ Also, $${\displaystyle \|x\|=\lim _{n\to \infty }\|s_{n}\|\leq \sum _{n=1}^{\infty }\|x_{n}\|<2L.}$$ This shows that ${\displaystyle y}$ belongs to ${\displaystyle A(2LU),}$ so ${\displaystyle V\subseteq A(2LU)}$ as claimed. Thus the image ${\displaystyle A(U)}$ of the unit ball in ${\displaystyle X}$ contains the open ball ${\displaystyle V/2L}$ of ${\displaystyle Y.}$ Hence, ${\displaystyle A(U)}$ is a neighborhood of the origin in ${\displaystyle Y,}$ and this concludes the proof.

Generalizations

Local convexity of ${\displaystyle X}$ or ${\displaystyle Y}$  is not essential to the proof, but completeness is: the theorem remains true in the case when ${\displaystyle X}$ and ${\displaystyle Y}$ are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:

Furthermore, in this latter case if ${\displaystyle N}$ is the kernel of ${\displaystyle A,}$ then there is a canonical factorization of ${\displaystyle A}$ in the form

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

where ${\displaystyle X/N}$ is the quotient space (also an F-space) of ${\displaystyle X}$ by the closed subspace ${\displaystyle N.}$ The quotient mapping ${\displaystyle X\to X/N}$ is open, and the mapping ${\displaystyle \alpha }$ is an isomorphism of topological vector spaces.^[7]

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

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

Nearly/Almost open linear maps

A linear map ${\displaystyle A:X\to Y}$ between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood ${\displaystyle U}$ of the origin in the domain, the closure of its image ${\displaystyle \operatorname {cl} A(U)}$ is a neighborhood of the origin in ${\displaystyle Y.}$^[9] Many authors use a different definition of "nearly/almost open map" that requires that the closure of ${\displaystyle A(U)}$ be a neighborhood of the origin in ${\displaystyle A(X)}$ rather than in ${\displaystyle Y,}$^[9] but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous.^[9] Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.^[10] The same is true of every surjective linear map from a TVS onto a Baire TVS.^[10]

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
• 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

References

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