Open mapping theorem (functional analysis)

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 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.

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]

• 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

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