Bipolar theorem

<h2 id="preliminaries">Preliminaries</h2>
<p class="note">Main article: <a href="/facts/Polar_set/JY45pfbt">Polar set</a></p>
<p>Suppose that 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) with a <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> 
  
    
      
        
          X
          
            ′
          
        
      
    
    {\displaystyle X^{\prime }}
  
 and let 
  
    
      
        
          ⟨
          
            x
            ,
            
              x
              
                ′
              
            
          
          ⟩
        
        :=
        
          x
          
            ′
          
        
        (
        x
        )
      
    
    {\displaystyle \left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x)}
  
 for all 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  
 and 
  
    
      
        
          x
          
            ′
          
        
        ∈
        
          X
          
            ′
          
        
        .
      
    
    {\displaystyle x^{\prime }\in X^{\prime }.}
  
 
The <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of a set 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 denoted by 
  
    
      
        co
        ⁡
        A
        ,
      
    
    {\displaystyle \operatorname {co} A,}
  
 is the smallest <a href="/facts/Convex_set/vdAuJRJl">convex set</a> containing 
  
    
      
        A
        .
      
    
    {\displaystyle A.}
  
 
The <a href="/facts/Convex_balanced_hull/rJ5HLWkg">convex balanced hull</a> of a set 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the smallest <a href="/facts/Convex_set/vdAuJRJl">convex</a> <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> set containing 
  
    
      
        A
        .
      
    
    {\displaystyle A.}

</p><p>The <a href="/facts/Polar_set/JY45pfbt">polar</a> of a subset 
  
    
      
        A
        ⊆
        X
      
    
    {\displaystyle A\subseteq X}
  
 is defined to be:

A
          
            ∘
          
        
        :=
        
          {
          
            
              x
              
                ′
              
            
            ∈
            
              X
              
                ′
              
            
            :
            
              sup
              
                a
                ∈
                A
              
            
            
              |
              
                ⟨
                
                  a
                  ,
                  
                    x
                    
                      ′
                    
                  
                
                ⟩
              
              |
            
            ≤
            1
          
          }
        
        .
      
    
    {\displaystyle A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}.}
  
 
while the prepolar of a subset 
  
    
      
        B
        ⊆
        
          X
          
            ′
          
        
      
    
    {\displaystyle B\subseteq X^{\prime }}
  
 is:

∘
          
        
        B
        :=
        
          {
          
            x
            ∈
            X
            :
            
              sup
              
                
                  x
                  
                    ′
                  
                
                ∈
                B
              
            
            
              |
              
                ⟨
                
                  x
                  ,
                  
                    x
                    
                      ′
                    
                  
                
                ⟩
              
              |
            
            ≤
            1
          
          }
        
        .
      
    
    {\displaystyle {}^{\circ }B:=\left\{x\in X:\sup _{x^{\prime }\in B}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.}

The bipolar of a subset 
  
    
      
        A
        ⊆
        X
        ,
      
    
    {\displaystyle A\subseteq X,}
  
 often denoted by 
  
    
      
        
          A
          
            ∘
            ∘
          
        
      
    
    {\displaystyle A^{\circ \circ }}
  
 is the set

A
          
            ∘
            ∘
          
        
        :=

∘
          
        
        
          (
          
            A
            
              ∘
            
          
          )
        
        =
        
          {
          
            x
            ∈
            X
            :
            
              sup
              
                
                  x
                  
                    ′
                  
                
                ∈
                
                  A
                  
                    ∘
                  
                
              
            
            
              |
              
                ⟨
                
                  x
                  ,
                  
                    x
                    
                      ′
                    
                  
                
                ⟩
              
              |
            
            ≤
            1
          
          }
        
        .
      
    
    {\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.}

</p>
<h2 id="statement-in-functional-analysis">Statement in functional analysis</h2>
<p>Let 
  
    
      
        σ
        
          (
          
            X
            ,
            
              X
              
                ′
              
            
          
          )
        
      
    
    {\displaystyle \sigma \left(X,X^{\prime }\right)}
  
 denote the <a href="/facts/Weak_topology/XJASE9Up">weak topology</a> on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (that is, the weakest TVS topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 making all linear functionals in 
  
    
      
        
          X
          
            ′
          
        
      
    
    {\displaystyle X^{\prime }}
  
 continuous).
</p>
The bipolar theorem:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The bipolar of a subset 
  
    
      
        A
        ⊆
        X
      
    
    {\displaystyle A\subseteq X}
  
 is equal to the 
  
    
      
        σ
        
          (
          
            X
            ,
            
              X
              
                ′
              
            
          
          )
        
      
    
    {\displaystyle \sigma \left(X,X^{\prime }\right)}
  
-closure of the <a href="/facts/Convex_balanced_hull/rJ5HLWkg">convex balanced hull</a> of 
  
    
      
        A
        .
      
    
    {\displaystyle A.}

<h2 id="statement-in-convex-analysis">Statement in convex analysis</h2>
The bipolar theorem:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: 54 <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> For any <a href="/facts/Nonempty_set/ffEk3eA6">nonempty</a> cone 
  
    
      
        A
      
    
    {\displaystyle A}
  
 in some <a href="/facts/Linear_space/M1pxMLx2">linear space</a> 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 the bipolar set 
  
    
      
        
          A
          
            ∘
            ∘
          
        
      
    
    {\displaystyle A^{\circ \circ }}
  
 is given by:
<p>
  
    
      
        
          A
          
            ∘
            ∘
          
        
        =
        cl
        ⁡
        (
        co
        ⁡
        {
        r
        a
        :
        r
        ≥
        0
        ,
        a
        ∈
        A
        }
        )
        .
      
    
    {\displaystyle A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).}

</p>
<h3>Special case</h3>
<p>A subset 
  
    
      
        C
        ⊆
        X
      
    
    {\displaystyle C\subseteq X}
  
 is a nonempty <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> <a href="/facts/Convex_cone/h1tARfw4">convex cone</a> if and only if 
  
    
      
        
          C
          
            +
            +
          
        
        =
        
          C
          
            ∘
            ∘
          
        
        =
        C
      
    
    {\displaystyle C^{++}=C^{\circ \circ }=C}
  
 when 
  
    
      
        
          C
          
            +
            +
          
        
        =
        
          
            (
            
              C
              
                +
              
            
            )
          
          
            +
          
        
        ,
      
    
    {\displaystyle C^{++}=\left(C^{+}\right)^{+},}
  
 where 
  
    
      
        
          A
          
            +
          
        
      
    
    {\displaystyle A^{+}}
  
 denotes the positive dual cone of a set 
  
    
      
        A
        .
      
    
    {\displaystyle A.}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
Or more generally, if 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is a nonempty convex cone then the bipolar cone is given by

C
          
            ∘
            ∘
          
        
        =
        cl
        ⁡
        C
        .
      
    
    {\displaystyle C^{\circ \circ }=\operatorname {cl} C.}

</p>
<h2 id="relation-to-the-fenchelmoreau-theorem">Relation to the <a href="/facts/Fenchel%25E2%2580%2593Moreau_theorem/PP3KnpNF">Fenchel–Moreau theorem</a></h2>
<p>Let

f
        (
        x
        )
        :=
        δ
        (
        x
        
          |
        
        C
        )
        =
        
          
            {
            
              
                
                  0
                
                
                  x
                  ∈
                  C
                
              
              
                
                  ∞
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle f(x):=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}}
  
 
be the <a href="/facts/Characteristic_function_(convex_analysis)/XfTwr4yd">indicator function</a> for a cone 
  
    
      
        C
        .
      
    
    {\displaystyle C.}

Then the <a href="/facts/Convex_conjugate/DGDWBrDL">convex conjugate</a>,

f
          
            ∗
          
        
        (
        
          x
          
            ∗
          
        
        )
        =
        δ
        
          (
          
            
              x
              
                ∗
              
            
            
              |
            
            
              C
              
                ∘
              
            
          
          )
        
        =
        
          δ
          
            ∗
          
        
        
          (
          
            
              x
              
                ∗
              
            
            
              |
            
            C
          
          )
        
        =
        
          sup
          
            x
            ∈
            C
          
        
        ⟨
        
          x
          
            ∗
          
        
        ,
        x
        ⟩
      
    
    {\displaystyle f^{*}(x^{*})=\delta \left(x^{*}|C^{\circ }\right)=\delta ^{*}\left(x^{*}|C\right)=\sup _{x\in C}\langle x^{*},x\rangle }
  
 
is the <a href="/facts/Support_function/W1k6H5lL">support function</a> for 
  
    
      
        C
        ,
      
    
    {\displaystyle C,}
  
 and 
  
    
      
        
          f
          
            ∗
            ∗
          
        
        (
        x
        )
        =
        δ
        (
        x
        
          |
        
        
          C
          
            ∘
            ∘
          
        
        )
        .
      
    
    {\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ }).}
  
 
Therefore, 
  
    
      
        C
        =
        
          C
          
            ∘
            ∘
          
        
      
    
    {\displaystyle C=C^{\circ \circ }}
  
 if and only if 
  
    
      
        f
        =
        
          f
          
            ∗
            ∗
          
        
        .
      
    
    {\displaystyle f=f^{**}.}
  
<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: 54 <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dual_system/2Y8KyofV">Dual system</a></li>
<li><a href="/facts/Fenchel%25E2%2580%2593Moreau_theorem/PP3KnpNF">Fenchel–Moreau theorem</a> – Mathematical theorem in convex analysis − A generalization of the bipolar theorem.</li>
<li><a href="/facts/Polar_set/JY45pfbt">Polar set</a> – Subset of all points that is bounded by some given point of a dual (in a dual pairing)</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. 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><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. 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></ul>

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

<ol>
<li id="fn:1"><p>Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. <a href="9780387295701" target="_blank">9780387295701</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Narici & Beckenstein 2011, pp. 225–273. - 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:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. <a href="9780387295701" target="_blank">9780387295701</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011. <a href="9780521833783" target="_blank">9780521833783</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011. <a href="9780521833783" target="_blank">9780521833783</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866. <a href="9780691015866" target="_blank">9780691015866</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. <a href="9780387295701" target="_blank">9780387295701</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866. <a href="9780691015866" target="_blank">9780691015866</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Bipolar theorem open-in-new

Bipolar theorem