Open mapping theorem (functional analysis)

<h2 id="statement-and-proof">Statement and proof</h2>

Open mapping theorem—<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Let 
 
 
 
 T
 :
 E
 →
 F
 
 
 {\displaystyle T:E\to F}
 
 be a surjective continuous linear map between Banach spaces (or more generally <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a>). 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 <a href="/facts/Baire_category_theorem/AwodvqgV">Baire category theorem</a>, and <a href="/facts/Complete_metric_space/lsckmU8G">completeness</a> 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 <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a>; 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 <a href="/facts/Nearly_open_map/ta3sk8AY">nearly open</a> 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—<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 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 <a href="/facts/Meager_set/AxTk8frw">non-meager</a> 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)—<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 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)—<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 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.

<h3>Transpose formulation</h3>
Here is a formulation of the open mapping theorem in terms of the <a href="/facts/Transpose/8wmsagGS">transpose</a> of an operator.

Theorem—<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
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 }
 
)

<ol><li>
 
 
 
 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}
 
;</li>
<li>
 
 
 
 δ
 
 B
 
 Y
 
 
 ⊂
 
 
 
 T
 
 (
 
 B
 
 X
 
 
 )
 
 
 ¯
 
 
 
 
 {\displaystyle \delta B_{Y}\subset {\overline {T\left(B_{X}\right)}}}
 
;</li>
<li>
 
 
 
 δ
 
 B
 
 Y
 
 
 ⊂
 
 T
 
 (
 
 B
 
 X
 
 
 )
 
 
 
 
 {\displaystyle \delta B_{Y}\subset {T\left(B_{X}\right)}}
 
;</li>
<li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 is surjective.</li></ol>
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 <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>. 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 <a href="/facts/Closed_range_theorem/ELWgtYFG">closed range theorem</a>, 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.

<h3>Quantative formulation</h3>
<a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a> gives the following quantitative formulation of the theorem:<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

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

<ol><li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 is open.</li>
<li>
 
 
 
 T
 
 
 {\displaystyle T}
 
 is surjective.</li>
<li>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\|}
 
.</li>
<li>3. holds for 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in some dense subspace of 
 
 
 
 F
 
 
 {\displaystyle F}
 
.</li></ol>

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 }

<h2 id="counterexample">Counterexample</h2>
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 <a href="/facts/Closed_graph_theorem_(functional_analysis)/l01mnNoJ">closed graph theorem</a>, a consequence of the open mapping theorem, fails without completeness. But here is a more concrete counterexample. Consider the space X of <a href="/facts/Sequence/gV2S4uqp">sequences</a> x : N → R with only finitely many non-zero terms equipped with the <a href="/facts/Supremum_norm/5yIkjNE3">supremum norm</a>. 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 <a href="/facts/Completeness_(topology)/lsckmU8G">complete</a>, 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 <a href="/facts/C_space/0bQSJKCE">space 
 
 
 
 
 c
 
 0
 
 
 
 
 {\displaystyle c_{0}}
 
</a> of all sequences that converge to zero, which is a (closed) subspace of the <a href="/facts/Lp_space/gbad0Rv1">ℓp space</a> 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}
 
.

<h2 id="consequences">Consequences</h2>
The open mapping theorem has several important consequences:

<ul><li>If 
 
 
 
 T
 :
 X
 →
 Y
 
 
 {\displaystyle T:X\to Y}
 
 is a <a href="/facts/Bijective/j4gTuTmW">bijective</a> continuous linear operator between the Banach spaces 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 then the <a href="/facts/Inverse_function/Tg5nnXlu">inverse operator</a> 
 
 
 
 
 T
 
 −
 1
 
 
 :
 Y
 →
 X
 
 
 {\displaystyle T^{-1}:Y\to X}
 
 is continuous as well (this is called the bounded inverse theorem).<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>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 <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 
 (
 
 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 <a href="/facts/Closed_graph_theorem_(functional_analysis)/l01mnNoJ">closed graph theorem</a>).<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li>
<li>Given a bounded operator 
 
 
 
 T
 :
 E
 →
 F
 
 
 {\displaystyle T:E\to F}
 
 between normed spaces, if the image of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is <a href="/facts/Non-meager/AxTk8frw">non-meager</a> 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).<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>An exact sequence of Banach spaces (or more generally Fréchet spaces) is topologically exact.</li>
<li>The <a href="/facts/Closed_range_theorem/ELWgtYFG">closed range theorem</a>, which says an operator (under some assumption) has closed image if and only if its transpose has closed image (see <a href="/facts/Closed_range_theorem/ELWgtYFG">closed range theorem#Sketch of proof</a>).</li></ul>
The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section. What we have is:<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<ul><li>A surjective continuous linear operator between Banach spaces admits a continuous linear section if and only if the kernel is topologically complemented.</li></ul>
In particular, the above applies to an operator between Hilbert spaces or an operator with finite-dimensional kernel (by the <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a>). 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.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="generalizations">Generalizations</h2>
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 <a href="/facts/F-space/NBCcYfcU">F-spaces</a>. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:

Open mapping theorem for continuous maps<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>—Let 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 be a <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operator</a> from a complete <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable TVS</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 onto a Hausdorff TVS 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}

If 
 
 
 
 Im
 ⁡
 A
 
 
 {\displaystyle \operatorname {Im} A}
 
 is <a href="/facts/Nonmeager_set/AxTk8frw">nonmeager</a> 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 <a href="/facts/F-space/NBCcYfcU">F-space</a>), 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 <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">kernel</a> 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 <a href="/facts/Quotient_space_(linear_algebra)/WmQ3lqdI">quotient space</a> (also an F-space) of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 by the <a href="/facts/Closed_set/upYnRTUj">closed</a> 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 <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> of <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
An important special case of this theorem can also be stated as

Theorem<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>—Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be two <a href="/facts/F-space/NBCcYfcU">F-spaces</a>. Then every continuous linear map of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 onto 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Topological_homomorphism/rnnTWdNf">TVS homomorphism</a>,
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 theorem<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>—Let 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 be a surjective <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> from a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable TVS</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 onto a TVS 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and suppose that at least one of the following two conditions is satisfied:

<ol><li>
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Baire_space/HToekdT8">Baire space</a>, or</li>
<li>
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is a <a href="/facts/Barrelled_space/r2H58TN3">barrelled space</a>,</li></ol>
If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Closed_linear_operator/l01mnNoJ">closed linear operator</a> then 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is an open mapping.
If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operator</a> 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 <a href="/facts/Almost_open_map/ta3sk8AY">almost open map</a>) 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.}
 
<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> 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,}
 
<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> but for surjective maps these definitions are equivalent.
A bijective linear map is nearly open if and only if its inverse is continuous.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
Every surjective linear map from <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex TVS</a> onto a <a href="/facts/Barrelled_space/r2H58TN3">barrelled TVS</a> is nearly open.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> The same is true of every surjective linear map from a TVS onto a <a href="/facts/Baire_space/HToekdT8">Baire</a> TVS.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>

Open mapping theorem<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>—If a closed surjective linear map from a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">pseudometrizable TVS</a> onto a Hausdorff TVS is nearly open then it is open.

Theorem<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>—If 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 is a continuous linear bijection from a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">Pseudometrizable</a> <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) onto a Hausdorff TVS that is a <a href="/facts/Baire_space/HToekdT8">Baire space</a>, then 
 
 
 
 A
 :
 X
 →
 Y
 
 
 {\displaystyle A:X\to Y}
 
 is a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> (and thus an isomorphism of TVSs).

<a href="/facts/Webbed_space/ChtDnm0X">Webbed spaces</a> are a class of <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> for which the open mapping theorem and the <a href="/facts/Closed_graph_theorem/CK2ra00W">closed graph theorem</a> hold.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Closed_graph/canolygI">Closed graph</a> – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Closed_graph_theorem/CK2ra00W">Closed graph theorem</a> – Theorem relating continuity to graphs</li>
<li><a href="/facts/Closed_graph_theorem_(functional_analysis)/l01mnNoJ">Closed graph theorem (functional analysis)</a> – Theorems connecting continuity to closure of graphs</li>
<li><a href="/facts/Open_mapping_theorem_(complex_analysis)/EtWbVUtu">Open mapping theorem (complex analysis)</a> – Theorem that holomorphic functions on complex domains are open mapsPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Surjection_of_Fr%25C3%25A9chet_spaces/06MelJYo">Surjection of Fréchet spaces</a> – Characterization of surjectivity</li>
<li><a href="/facts/Ursescu_theorem/4LsjFGo8">Ursescu theorem</a> – Generalization of closed graph, open mapping, and uniform boundedness theorem</li>
<li><a href="/facts/Webbed_space/ChtDnm0X">Webbed space</a> – Space where open mapping and closed graph theorems hold</li></ul>

<h2 id="bibliography">Bibliography</h2>
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<li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li>Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-29882-7. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/589250">589250</a>.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1973). <a href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780070542259.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li>
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<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li>
<li>Vogt, Dietmar (2000). <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf">"Lectures on Fréchet spaces"</a> (PDF). Bergische Universität Wuppertal.</li>
<li><a href="/facts/Albert_Wilansky/I6QRVtGa">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49353-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/849801114">849801114</a>.</li></ul>
This article incorporates material from Proof of open mapping theorem on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="https://mathoverflow.net/questions/383316/when-is-a-complex-of-banach-spaces-exact-as-condensed-abelian-groups">"When is a complex of Banach spaces exact as condensed abelian groups?"</a>. MathOverflow. February 6, 2021.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Vogt 2000, Theorem 1.6. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf" target="_blank">https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Vogt 2000, Lemma 1.4. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf" target="_blank">https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Vogt 2000, Lemma 1.5. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf" target="_blank">https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Vogt 2000, Corollary 1.7. - Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal. <a href="https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf" target="_blank">https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Tao, Terence (February 1, 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's New. <a href="/wiki/Terence_Tao" target="_blank">/wiki/Terence_Tao</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Tao, Terence (February 1, 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's New. <a href="/wiki/Terence_Tao" target="_blank">/wiki/Terence_Tao</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Sarnowski, Jarek (October 31, 2020). "Can the inverse operator in Bartle-Graves theorem be linear?". MathOverflow. <a href="https://mathoverflow.net/questions/375326/can-the-inverse-operator-in-bartle-graves-theorem-be-linear" target="_blank">https://mathoverflow.net/questions/375326/can-the-inverse-operator-in-bartle-graves-theorem-be-linear</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Jonathan_Borwein" target="_blank">/wiki/Jonathan_Borwein</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://archive.org/details/functionalanalys00rudi" target="_blank">https://archive.org/details/functionalanalys00rudi</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Dieudonné 1970, 12.16.8. - Dieudonné, Jean (1970). Treatise on Analysis, Volume II. Academic Press. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
</ol>

Open mapping theorem (functional analysis) open-in-new

Open mapping theorem (functional analysis)