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Preliminaries

Main article: Polar set

Suppose that X {\displaystyle X} is a topological vector space (TVS) with a continuous dual space 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 convex hull of a set A , {\displaystyle A,} denoted by co ⁡ A , {\displaystyle \operatorname {co} A,} is the smallest convex set containing A . {\displaystyle A.} The convex balanced hull of a set A {\displaystyle A} is the smallest convex balanced set containing A . {\displaystyle A.}

The polar 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\}.}

Statement in functional analysis

Let σ ( X , X ′ ) {\displaystyle \sigma \left(X,X^{\prime }\right)} denote the weak topology on X {\displaystyle X} (that is, the weakest TVS topology on X {\displaystyle X} making all linear functionals in X ′ {\displaystyle X^{\prime }} continuous).

The bipolar theorem:² 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 convex balanced hull of A . {\displaystyle A.}

Statement in convex analysis

The bipolar theorem:³: 54 ⁴ For any nonempty cone A {\displaystyle A} in some linear space X , {\displaystyle X,} the bipolar set A ∘ ∘ {\displaystyle A^{\circ \circ }} is given by:

A ∘ ∘ = cl ⁡ ( co ⁡ { r a : r ≥ 0 , a ∈ A } ) . {\displaystyle A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).}

Special case

A subset C ⊆ X {\displaystyle C\subseteq X} is a nonempty closed convex cone 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.} ⁵⁶ 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.}

Relation to the Fenchel–Moreau theorem

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 indicator function for a cone C . {\displaystyle C.} Then the convex conjugate, 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 support function 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^{**}.} ⁷: 54 ⁸

See also

• Dual system
• Fenchel–Moreau theorem – Mathematical theorem in convex analysis − A generalization of the bipolar theorem.
• Polar set – Subset of all points that is bounded by some given point of a dual (in a dual pairing)

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

References

1. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. 9780387295701 ↩

2. 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. https://search.worldcat.org/oclc/144216834 ↩

3. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. 9780387295701 ↩

4. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011. 9780521833783 ↩

5. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011. 9780521833783 ↩

6. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866. 9780691015866 ↩

7. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. 9780387295701 ↩

8. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866. 9780691015866 ↩