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Statement and proof

Open mapping theorem—²³ Let T : E → F {\displaystyle T:E\to F} be a surjective continuous linear map between Banach spaces (or more generally Fréchet spaces). Then T {\displaystyle T} is an open mapping (that is, if U ⊂ E {\displaystyle U\subset E} is an open subset, then T ( U ) {\displaystyle T(U)} is open).

The proof here uses the Baire category theorem, and completeness of both E {\displaystyle E} and F {\displaystyle F} is essential to the theorem. The statement of the theorem is no longer true if either space is assumed to be only a normed vector space; see § Counterexample.

The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map f : E → F {\displaystyle f:E\to F} between topological vector spaces is said to be nearly open if, for each neighborhood U {\displaystyle U} of zero, the closure f ( U ) ¯ {\displaystyle {\overline {f(U)}}} contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.

Lemma—⁴⁵ A linear map f : E → F {\displaystyle f:E\to F} between normed spaces is nearly open if the image of f {\displaystyle f} is non-meager in F {\displaystyle F} . (The continuity is not needed.)

Proof: Shrinking U {\displaystyle U} , we can assume U {\displaystyle U} is an open ball centered at zero. We have f ( E ) = f ( ⋃ n ∈ N n U ) = ⋃ n ∈ N f ( n U ) {\displaystyle f(E)=f\left(\bigcup _{n\in \mathbb {N} }nU\right)=\bigcup _{n\in \mathbb {N} }f(nU)} . Thus, some f ( n U ) ¯ {\displaystyle {\overline {f(nU)}}} contains an interior point y {\displaystyle y} ; that is, for some radius r > 0 {\displaystyle r>0} ,

B ( y , r ) ⊂ f ( n U ) ¯ . {\displaystyle B(y,r)\subset {\overline {f(nU)}}.}

Then for any v {\displaystyle v} in F {\displaystyle F} with ‖ v ‖ < r {\displaystyle \|v\|<r} , by linearity, convexity and ( − 1 ) U ⊂ U {\displaystyle (-1)U\subset U} ,

v = v − y + y ∈ f ( − n U ) ¯ + f ( n U ) ¯ ⊂ f ( 2 n U ) ¯ {\displaystyle v=v-y+y\in {\overline {f(-nU)}}+{\overline {f(nU)}}\subset {\overline {f(2nU)}}} ,

which proves the lemma by dividing by 2 n {\displaystyle 2n} . ◻ {\displaystyle \square } (The same proof works if E , F {\displaystyle E,F} are pre-Fréchet spaces.)

The completeness on the domain then allows to upgrade nearly open to open.

Lemma (Schauder)—⁶⁷ Let f : E → F {\displaystyle f:E\to F} be a continuous linear map between normed spaces.

If f {\displaystyle f} is nearly-open and if E {\displaystyle E} is complete, then f {\displaystyle f} is open and surjective.

More precisely, if B ( 0 , δ ) ⊂ f ( B ( 0 , 1 ) ) ¯ {\displaystyle B(0,\delta )\subset {\overline {f(B(0,1))}}} for some δ > 0 {\displaystyle \delta >0} and if E {\displaystyle E} is complete, then

B ( 0 , δ ) ⊂ f ( B ( 0 , 1 ) ) {\displaystyle B(0,\delta )\subset f(B(0,1))}

where B ( x , r ) {\displaystyle B(x,r)} is an open ball with radius r {\displaystyle r} and center x {\displaystyle x} .

Proof: Let y {\displaystyle y} be in B ( 0 , δ ) {\displaystyle B(0,\delta )} and c n > 0 {\displaystyle c_{n}>0} some sequence. We have: B ( 0 , δ ) ¯ ⊂ f ( B ( 0 , 1 ) ) ¯ {\displaystyle {\overline {B(0,\delta )}}\subset {\overline {f(B(0,1))}}} . Thus, for each ϵ > 0 {\displaystyle \epsilon >0} and z {\displaystyle z} in F {\displaystyle F} , we can find an x {\displaystyle x} with ‖ x ‖ < δ − 1 ‖ z ‖ {\displaystyle \|x\|<\delta ^{-1}\|z\|} and z {\displaystyle z} in B ( f ( x ) , ϵ ) {\displaystyle B(f(x),\epsilon )} . Thus, taking z = y {\displaystyle z=y} , we find an x 1 {\displaystyle x_{1}} such that

‖ y − f ( x 1 ) ‖ < c 1 , ‖ x 1 ‖ < δ − 1 ‖ y ‖ . {\displaystyle \|y-f(x_{1})\|<c_{1},\,\|x_{1}\|<\delta ^{-1}\|y\|.}

Applying the same argument with z = y − f ( x 1 ) {\displaystyle z=y-f(x_{1})} , we then find an x 2 {\displaystyle x_{2}} such that

‖ y − f ( x 1 ) − f ( x 2 ) ‖ < c 2 , ‖ x 2 ‖ < δ − 1 c 1 {\displaystyle \|y-f(x_{1})-f(x_{2})\|<c_{2},\,\|x_{2}\|<\delta ^{-1}c_{1}}

where we observed ‖ x 2 ‖ < δ − 1 ‖ z ‖ < δ − 1 c 1 {\displaystyle \|x_{2}\|<\delta ^{-1}\|z\|<\delta ^{-1}c_{1}} . Then so on. Thus, if c := ∑ c n < ∞ {\displaystyle c:=\sum c_{n}<\infty } , we found a sequence x n {\displaystyle x_{n}} such that x = ∑ 1 ∞ x n {\displaystyle x=\sum _{1}^{\infty }x_{n}} converges and f ( x ) = y {\displaystyle f(x)=y} . Also,

‖ x ‖ ≤ ∑ 1 ∞ ‖ x n ‖ ≤ δ − 1 ‖ y ‖ + δ − 1 c . {\displaystyle \|x\|\leq \sum _{1}^{\infty }\|x_{n}\|\leq \delta ^{-1}\|y\|+\delta ^{-1}c.}

Since δ − 1 ‖ y ‖ < 1 {\displaystyle \delta ^{-1}\|y\|<1} , by making c {\displaystyle c} small enough, we can achieve ‖ x ‖ < 1 {\displaystyle \|x\|<1} . ◻ {\displaystyle \square } (Again the same proof is valid if E , F {\displaystyle E,F} are pre-Fréchet spaces.)

Proof of the theorem: By Baire's category theorem, the first lemma applies. Then the conclusion of the theorem follows from the second lemma. ◻ {\displaystyle \square }

In general, a continuous bijection between topological spaces is not necessarily a homeomorphism. The open mapping theorem, when it applies, implies the bijectivity is enough:

Corollary (Bounded inverse theorem)—⁸ A continuous bijective linear operator between Banach spaces (or Fréchet spaces) has continuous inverse. That is, the inverse operator is continuous.

Even though the above bounded inverse theorem is a special case of the open mapping theorem, the open mapping theorem in turns follows from that. Indeed, a surjective continuous linear operator T : E → F {\displaystyle T:E\to F} factors as

T : E → p E / ker ⁡ T → T 0 F . {\displaystyle T:E{\overset {p}{\to }}E/\operatorname {ker} T{\overset {T_{0}}{\to }}F.}

Here, T 0 {\displaystyle T_{0}} is continuous and bijective and thus is a homeomorphism by the bounded inverse theorem; in particular, it is an open mapping. As a quotient map for topological groups is open, T {\displaystyle T} is open then.

Because the open mapping theorem and the bounded inverse theorem are essentially the same result, they are often simply called Banach's theorem.

Transpose formulation

Here is a formulation of the open mapping theorem in terms of the transpose of an operator.

Theorem—⁹ Let X {\displaystyle X} and Y {\displaystyle Y} be Banach spaces, let B X {\displaystyle B_{X}} and B Y {\displaystyle B_{Y}} denote their open unit balls, and let T : X → Y {\displaystyle T:X\to Y} be a bounded linear operator. If δ > 0 {\displaystyle \delta >0} then among the following four statements we have ( 1 ) ⟹ ( 2 ) ⟹ ( 3 ) ⟹ ( 4 ) {\displaystyle (1)\implies (2)\implies (3)\implies (4)} (with the same δ {\displaystyle \delta } )

1. δ ‖ y ′ ‖ ≤ ‖ T ′ y ′ ‖ {\displaystyle \delta \left\|y'\right\|\leq \left\|T'y'\right\|} for all y ′ ∈ Y ′ {\displaystyle y'\in Y'} = continuous dual of Y {\displaystyle Y} ;
2. δ B Y ⊂ T ( B X ) ¯ {\displaystyle \delta B_{Y}\subset {\overline {T\left(B_{X}\right)}}} ;
3. δ B Y ⊂ T ( B X ) {\displaystyle \delta B_{Y}\subset {T\left(B_{X}\right)}} ;
4. T {\displaystyle T} is surjective.

Furthermore, if T {\displaystyle T} is surjective then (1) holds for some δ > 0. {\displaystyle \delta >0.}

Proof: The idea of 1. ⇒ {\displaystyle \Rightarrow } 2. is to show: y ∉ T ( B X ) ¯ ⇒ ‖ y ‖ > δ , {\displaystyle y\notin {\overline {T(B_{X})}}\Rightarrow \|y\|>\delta ,} and that follows from the Hahn–Banach theorem. 2. ⇒ {\displaystyle \Rightarrow } 3. is exactly the second lemma in § Statement and proof. Finally, 3. ⇒ {\displaystyle \Rightarrow } 4. is trivial and 4. ⇒ {\displaystyle \Rightarrow } 1. easily follows from the open mapping theorem. ◻ {\displaystyle \square }

Alternatively, 1. implies that T ′ {\displaystyle T'} is injective and has closed image and then by the closed range theorem, that implies T {\displaystyle T} has dense image and closed image, respectively; i.e., T {\displaystyle T} is surjective. Hence, the above result is a variant of a special case of the closed range theorem.

Quantative formulation

Terence Tao gives the following quantitative formulation of the theorem:¹⁰

Theorem—Let T : E → F {\displaystyle T:E\to F} be a bounded operator between Banach spaces. Then the following are equivalent:

1. T {\displaystyle T} is open.
2. T {\displaystyle T} is surjective.
3. There exists a constant C > 0 {\displaystyle C>0} such that, for each f {\displaystyle f} in F {\displaystyle F} , the equation T u = f {\displaystyle Tu=f} has a solution u {\displaystyle u} with ‖ u ‖ ≤ C ‖ f ‖ {\displaystyle \|u\|\leq C\|f\|} .
4. 3. holds for f {\displaystyle f} in some dense subspace of F {\displaystyle F} .

Proof: 2. ⇒ {\displaystyle \Rightarrow } 1. is the usual open mapping theorem.

1. ⇒ {\displaystyle \Rightarrow } 4.: For some r > 0 {\displaystyle r>0} , we have B ( 0 , 2 ) ⊂ T ( B ( 0 , r ) ) {\displaystyle B(0,2)\subset T(B(0,r))} where B {\displaystyle B} means an open ball. Then f ‖ f ‖ = T ( u ‖ f ‖ ) {\displaystyle {\frac {f}{\|f\|}}=T\left({\frac {u}{\|f\|}}\right)} for some u ‖ f ‖ {\displaystyle {\frac {u}{\|f\|}}} in B ( 0 , r ) {\displaystyle B(0,r)} . That is, T u = f {\displaystyle Tu=f} with ‖ u ‖ < r ‖ f ‖ {\displaystyle \|u\|<r\|f\|} .

4. ⇒ {\displaystyle \Rightarrow } 3.: We can write f = ∑ 0 ∞ f j {\displaystyle f=\sum _{0}^{\infty }f_{j}} with f j {\displaystyle f_{j}} in the dense subspace and the sum converging in norm. Then, since E {\displaystyle E} is complete, u = ∑ 0 ∞ u j {\displaystyle u=\sum _{0}^{\infty }u_{j}} with ‖ u j ‖ ≤ C ‖ f j ‖ {\displaystyle \|u_{j}\|\leq C\|f_{j}\|} and T u j = f j {\displaystyle Tu_{j}=f_{j}} is a required solution. Finally, 3. ⇒ {\displaystyle \Rightarrow } 2. is trivial. ◻ {\displaystyle \square }

Counterexample

The open mapping theorem may not hold for normed spaces that are not complete. A quickest way to see this is to note that the closed graph theorem, a consequence of the open mapping theorem, fails without completeness. But here is a more concrete counterexample. Consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm. The map T : X → X defined by

T x = ( x 1 , x 2 2 , x 3 3 , … ) {\displaystyle Tx=\left(x_{1},{\frac {x_{2}}{2}},{\frac {x_{3}}{3}},\dots \right)}

is bounded, linear and invertible, but T−1 is unbounded. This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences x(n) ∈ X given by

x ( n ) = ( 1 , 1 2 , … , 1 n , 0 , 0 , … ) {\displaystyle x^{(n)}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},0,0,\dots \right)}

converges as n → ∞ to the sequence x(∞) given by

x ( ∞ ) = ( 1 , 1 2 , … , 1 n , … ) , {\displaystyle x^{(\infty )}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},\dots \right),}

which has all its terms non-zero, and so does not lie in X.

The completion of X is the space c 0 {\displaystyle c_{0}} of all sequences that converge to zero, which is a (closed) subspace of the ℓp space ℓ∞(N), which is the space of all bounded sequences. However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence

x = ( 1 , 1 2 , 1 3 , … ) , {\displaystyle x=\left(1,{\frac {1}{2}},{\frac {1}{3}},\dots \right),}

is an element of c 0 {\displaystyle c_{0}} , but is not in the range of T : c 0 → c 0 {\displaystyle T:c_{0}\to c_{0}} . Same reasoning applies to show T {\displaystyle T} is also not onto in l ∞ {\displaystyle l^{\infty }} , for example x = ( 1 , 1 , 1 , … ) {\displaystyle x=\left(1,1,1,\dots \right)} is not in the range of T {\displaystyle T} .

Consequences

The open mapping theorem has several important consequences:

• If T : X → Y {\displaystyle T:X\to Y} is a bijective continuous linear operator between the Banach spaces X {\displaystyle X} and Y , {\displaystyle Y,} then the inverse operator T − 1 : Y → X {\displaystyle T^{-1}:Y\to X} is continuous as well (this is called the bounded inverse theorem).¹¹
• If T : X → Y {\displaystyle T:X\to Y} is a linear operator between the Banach spaces X {\displaystyle X} and Y , {\displaystyle Y,} and if for every sequence ( x n ) {\displaystyle \left(x_{n}\right)} in X {\displaystyle X} with x n → 0 {\displaystyle x_{n}\to 0} and T x n → y {\displaystyle Tx_{n}\to y} it follows that y = 0 , {\displaystyle y=0,} then T {\displaystyle T} is continuous (the closed graph theorem).¹²
• Given a bounded operator T : E → F {\displaystyle T:E\to F} between normed spaces, if the image of T {\displaystyle T} is non-meager and if E {\displaystyle E} is complete, then T {\displaystyle T} is open and surjective and F {\displaystyle F} is complete (to see this, use the two lemmas in the proof of the theorem).¹³
• An exact sequence of Banach spaces (or more generally Fréchet spaces) is topologically exact.
• The closed range theorem, which says an operator (under some assumption) has closed image if and only if its transpose has closed image (see closed range theorem#Sketch of proof).

The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section. What we have is:¹⁴

• A surjective continuous linear operator between Banach spaces admits a continuous linear section if and only if the kernel is topologically complemented.

In particular, the above applies to an operator between Hilbert spaces or an operator with finite-dimensional kernel (by the Hahn–Banach theorem). If one drops the requirement that a section be linear, a surjective continuous linear operator between Banach spaces admits a continuous section; this is the Bartle–Graves theorem.¹⁵¹⁶

Generalizations

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

Open mapping theorem for continuous maps¹⁷¹⁸—Let A : X → Y {\displaystyle A:X\to Y} be a continuous linear operator from a complete pseudometrizable TVS X {\displaystyle X} onto a Hausdorff TVS Y . {\displaystyle Y.} If Im ⁡ A {\displaystyle \operatorname {Im} A} is nonmeager in Y {\displaystyle Y} then A : X → Y {\displaystyle A:X\to Y} is a (surjective) open map and Y {\displaystyle Y} is a complete pseudometrizable TVS. Moreover, if X {\displaystyle X} is assumed to be hausdorff (i.e. a F-space), then Y {\displaystyle Y} is also an F-space.

(The proof is essentially the same as the Banach or Fréchet cases; we modify the proof slightly to avoid the use of convexity,)

Furthermore, in this latter case if N {\displaystyle N} is the kernel of A , {\displaystyle A,} then there is a canonical factorization of A {\displaystyle A} in the form X → X / N → α Y {\displaystyle X\to X/N{\overset {\alpha }{\to }}Y} where X / N {\displaystyle X/N} is the quotient space (also an F-space) of X {\displaystyle X} by the closed subspace N . {\displaystyle N.} The quotient mapping X → X / N {\displaystyle X\to X/N} is open, and the mapping α {\displaystyle \alpha } is an isomorphism of topological vector spaces.¹⁹

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

Theorem²⁰—Let X {\displaystyle X} and Y {\displaystyle Y} be two F-spaces. Then every continuous linear map of X {\displaystyle X} onto Y {\displaystyle Y} is a TVS homomorphism, where a linear map u : X → Y {\displaystyle u:X\to Y} is a topological vector space (TVS) homomorphism if the induced map u ^ : X / ker ⁡ ( u ) → Y {\displaystyle {\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 theorem²¹—Let A : X → Y {\displaystyle A:X\to Y} be a surjective linear map from a complete pseudometrizable TVS X {\displaystyle X} onto a TVS Y {\displaystyle Y} and suppose that at least one of the following two conditions is satisfied:

1. Y {\displaystyle Y} is a Baire space, or
2. X {\displaystyle X} is locally convex and Y {\displaystyle Y} is a barrelled space,

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

Nearly/Almost open linear maps

A linear map A : X → Y {\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 U {\displaystyle U} of the origin in the domain, the closure of its image cl ⁡ A ( U ) {\displaystyle \operatorname {cl} A(U)} is a neighborhood of the origin in Y . {\displaystyle Y.} ²² Many authors use a different definition of "nearly/almost open map" that requires that the closure of A ( U ) {\displaystyle A(U)} be a neighborhood of the origin in A ( X ) {\displaystyle A(X)} rather than in Y , {\displaystyle Y,} ²³ but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous.²⁴ Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.²⁵ The same is true of every surjective linear map from a TVS onto a Baire TVS.²⁶

Open mapping theorem²⁷—If a closed surjective linear map from a complete pseudometrizable TVS onto a Hausdorff TVS is nearly open then it is open.

Theorem²⁸—If A : X → Y {\displaystyle 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 → Y {\displaystyle A:X\to Y} is a homeomorphism (and thus an isomorphism of TVSs).

Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.

See also

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

Further reading

• "When is a complex of Banach spaces exact as condensed abelian groups?". MathOverflow. February 6, 2021.

References

1. Trèves 2006, p. 166. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

2. Rudin 1973, Theorem 2.11. - Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. https://archive.org/details/functionalanalys00rudi ↩

3. Vogt 2000, Theorem 1.6. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf ↩

4. Vogt 2000, Lemma 1.4. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf ↩

5. The first part of the proof of Rudin 1991, Theorem 2.11. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

6. Rudin 1991, Theorem 4.13. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

7. Vogt 2000, Lemma 1.5. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf ↩

8. Vogt 2000, Corollary 1.7. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf ↩

9. Rudin 1991, Theorem 4.13. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

10. Tao, Terence (February 1, 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's New. /wiki/Terence_Tao ↩

11. Rudin 1973, Corollary 2.12. - Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. https://archive.org/details/functionalanalys00rudi ↩

12. Rudin 1973, Theorem 2.15. - Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. https://archive.org/details/functionalanalys00rudi ↩

13. Rudin 1991, Theorem 2.11. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

14. Tao, Terence (February 1, 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's New. /wiki/Terence_Tao ↩

15. Sarnowski, Jarek (October 31, 2020). "Can the inverse operator in Bartle-Graves theorem be linear?". MathOverflow. https://mathoverflow.net/questions/375326/can-the-inverse-operator-in-bartle-graves-theorem-be-linear ↩

16. Borwein, J. M.; Dontchev, A. L. (2003). "On the Bartle–Graves theorem". Proceedings of the American Mathematical Society. 131 (8): 2553–2560. doi:10.1090/S0002-9939-03-07229-0. hdl:1959.13/940334. MR 1974655. /wiki/Jonathan_Borwein ↩

17. Rudin 1991, Theorem 2.11. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

18. Narici & Beckenstein 2011, p. 468. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

19. Dieudonné 1970, 12.16.8. - Dieudonné, Jean (1970). Treatise on Analysis, Volume II. Academic Press. ↩

20. Trèves 2006, p. 170 - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

21. Narici & Beckenstein 2011, p. 468. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

22. Narici & Beckenstein 2011, pp. 466. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

23. Narici & Beckenstein 2011, pp. 466. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

24. Narici & Beckenstein 2011, pp. 466. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

25. Narici & Beckenstein 2011, pp. 467. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

26. Narici & Beckenstein 2011, pp. 467. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

27. Narici & Beckenstein 2011, pp. 466−468. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

28. Narici & Beckenstein 2011, p. 469. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩