Polar set

In functional and convex analysis, related disciplines of mathematics, the polar set A° is a special convex set associated to any subset A of a vector space X lying in the dual space ${\displaystyle X^{\prime }}$. The bipolar of a subset is the polar of A°, but lies in X (not ${\displaystyle X^{\prime \prime }}$).

See also polar set (potential theory).

Definitions

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.[1] In each case, the definition describes a duality between certain subsets of a pairing of vector spaces ⟨X, Y⟩ over the real or complex numbers (X and Y are often topological vector spaces (TVSs)).

Absolute polar

Suppose that ${\displaystyle \left\langle X,Y\right\rangle }$ is a pairing. The polar or absolute polar of a subset A of X is the set:

${\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{a\in A}\left|\left\langle a,y\right\rangle \right|\leq 1\right\}=\left\{y\in Y:\sup \left|\left\langle A,y\right\rangle \right|\leq 1\right\}}$

where ${\displaystyle \left|\left\langle A,y\right\rangle \right|:=\left\{\left|\left\langle a,y\right\rangle \right|:a\in A\right\}}$.

This is an affine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball (in X) is precisely the unit ball (in Y).

The prepolar or absolute prepolar of a subset B of Y is the set:

${\displaystyle {}^{\circ }B:=\left\{x\in X:\sup _{b\in B}\left|\left\langle x,b\right\rangle \right|\leq 1\right\}=\left\{x\in X:\sup \left|\left\langle x,B\right\rangle \right|\leq 1\right\}}$

Very often, the prepolar of a subset B of Y is also called the polar or absolute polar of B and denoted by B°; in practice, this reuse of notation and of the word "polar" rarely causes any issues (such as ambiguity) and many authors do not even use the word "prepolar".

The bipolar of a subset A of X, often denoted by A°°, is the set ${\displaystyle {}^{\circ }\left(A^{\circ }\right)}$; that is,

${\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{y\in A^{\circ }}\left|\left\langle x,y\right\rangle \right|\leq 1\right\}}$.

Real polar

The real polar of a subset A of X is the set:

${\displaystyle A^{r}:=\left\{y\in Y:\sup _{a\in A}\operatorname {Re} \left\langle a,y\right\rangle \leq 1\right\}}$

and the real prepolar of a subset B of Y is the set:

${\displaystyle {}^{r}B:=\left\{x\in X:\sup _{b\in B}\operatorname {Re} \left\langle x,b\right\rangle \leq 1\right\}}$.

As with the absolute prepolar, the real prepolar is usually called the real polar and is also denoted by B^r.[2] It's important to note that some authors (e.g. [Schaefer 1999]) define "polar" to mean "real polar" (rather than "absolute polar", as is done in this article) and use the notation A° for it (rather than the notation A^r that is used in this article and in [Narici 2011]).

The real bipolar of a subset A of X, sometimes denoted by A^{r r}, is the set ${\displaystyle {}^{r}\left(A^{r}\right)}$; it is equal to the 𝜎(X, Y)-closure of the convex hull of A ∪ { 0 }.[2]

For a subset A of X, A^r is convex, 𝜎(Y, X)-closed, and contains A°.[2] In general, it is possible that A° ≠ A^r but equality will hold if A is balanced. Furthermore, A^∘ = ${\displaystyle \left(\operatorname {bal} \left(A^{r}\right)\right)}$ where bal(A^r) denotes the balanced hull of A^r.[2]

Competing definitions

The definition of the "polar" of a set is not universally agreed upon. Although this article defined "polar" to mean "absolute polar", some authors define "polar" to mean "real polar" and other authors use still other definitions. No matter how an author defines "polar", the notation A° almost always represents their choice of the definition (so the meaning of the notation A° may vary from source to source). In particular, the polar of A is sometimes defined as:

${\displaystyle A^{|r|}:=\left\{y\in Y:\sup _{a\in A}\left|\operatorname {Re} \left\langle a,y\right\rangle \right|\leq 1\right\}}$

where the notation A^|r| is not standard notation.

We now briefly discuss how these various definitions relate to one another and when they are equivalent.

We always have A° ⊆ A^|r| ⊆ A^r and if ⟨•, •⟩ is real-valued (or equivalently, if X and Y are vector spaces over ℝ) then A° = A^|r|.

If A is symmetric (i.e. -A = A or equivalently, -A ⊆ A) then A^|r| = A^r where if in addition ⟨•, •⟩ is real-valued then A° = A^|r| = A^r.

If X and Y are vector spaces over ℂ (so that ⟨•, •⟩ is complex-valued) and if iA ⊆ A (where note that this implies -A = A and iA = A), then

A° ⊆ A^|r| = A^r ⊆ (1/√2 A)°

where if in addition ${\displaystyle e^{ir}}$A ⊆ A for all real r then A^∘ = A^r.[2]

Thus for all of these definitions of the polar set of A to agree, it suffices that sA ⊆ A for all scalars s of unit length[nb 1] (where this is equivalent to sA = A for all unit length scalar s). In particular, all definitions of the polar of A agree when A is a balanced set (which is often, but not always, the case) so that often, which of these competing definitions is used is immaterial. However, these difference in the definitions of the "polar" of a set A do sometimes introduce subtle or important technical differences when A is not necessarily balanced.

Specialization for the canonical duality

Suppose that X is a topological vector space (TVS) with continuous dual space ${\displaystyle X^{\prime }}$. We consider the important special case where Y := ${\displaystyle X^{\prime }}$ and the brackets represent the canonical map (i.e. ${\displaystyle \left\langle x,x^{\prime }\right\rangle$ :=x^{\prime }(x)} ). Thus ${\displaystyle \left\langle X,X^{\prime }\right\rangle }$ is the canonical pairing.

The polar of a subset A ⊆ X is:

${\displaystyle A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|x^{\prime }(a)\right|\leq 1\right\}=\left\{x^{\prime }\in X^{\prime }:\sup \left|x^{\prime }(A)\right|\leq 1\right\}}$

If A satisfies sA ⊆ A for all scalars s of unit length then one may replace the absolute value signs by ${\displaystyle \operatorname {Re} }$ (the real part operator) so that:

${\displaystyle A^{\circ }=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\operatorname {Re} x^{\prime }(a)\leq 1\right\}=\left\{x^{\prime }\in X^{\prime }:\sup \operatorname {Re} x^{\prime }(A)\leq 1\right\}}$.

The prepolar of a subset B of ${\displaystyle Y=X^{\prime }}$ is:

${\displaystyle {}^{\circ }B:=\left\{x\in X:\sup _{b^{\prime }\in B}\left|b^{\prime }(x)\right|\leq 1\right\}=\left\{x\in X:\sup \left|B(x)\right|\leq 1\right\}}$

If B satisfies sB ⊆ B for all scalars s of unit length then one may replace the absolute value signs with ${\displaystyle \operatorname {Re} }$ so that:

${\displaystyle {}^{\circ }B=\left\{x\in X:\sup _{b^{\prime }\in B}\operatorname {Re} b^{\prime }(x)\leq 1\right\}=\left\{x\in X:\sup \operatorname {Re} B(x)\leq 1\right\}}$

where B(x) := ${\displaystyle \left\{b^{\prime }(x):b^{\prime }\in B\right\}}$.

The bipolar theorem characterizes the bipolar of a subset of a topological vector space.

If X is a normed space and S is the open or closed unit ball in X (or even any subset of the closed unit ball that contains the open unit ball) then S^∘ is the closed unit ball in the continuous dual space ${\displaystyle X^{\prime }}$ when ${\displaystyle X^{\prime }}$ is endowed with its canonical dual norm.

Geometric definition for cones

The polar cone of a convex cone A ⊆ X is the set

${\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{x\in A}\langle x,y\rangle \leq 0\right\}}$

This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x ∈ X is the locus ${\displaystyle \{y:\langle y,x\rangle =0\}}$; the dual relationship for a hyperplane yields that hyperplane's polar point.[3]

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.[4]

Properties

Unless stated otherwise, we henceforth assume that ⟨X, Y⟩ is a pairing. Note that 𝜎(Y, X) is the weak-* topology on Y while 𝜎(X, Y) is the weak topology on X. For any set A}, A^r denotes the real polar of A and A° denotes the absolute polar of A. By "polar" we mean the absolute polar.

• The (absolute) polar of a set is convex and balanced.[5]
• The real polar A^r of a subset A of X is convex but not necessarily balanced; A^r will be balanced if A is balanced.[6]
• If sA ⊆ A for all scalars s of unit length then A° = A^r.
• A° is closed in Y under the weak-*-topology on Y.[3]
• A subset S of X is weakly bounded (i.e. 𝜎(X, Y)-bounded) if and only if S° is absorbing in Y.[2]
• For a dual pair (X, X'), where X is a TVS and X' is its continuous dual space, if B ⊆ X is bounded then B° is absorbing in X'.[5] If X is locally convex and B° is absorbing in X' then B is bounded in X. Moreover, a subset S of X is weakly bounded if and only if S° is absorbing in X'.
• The bipolar A°° of a set A is the 𝜎(X, Y)-closed convex hull of A ∪ { 0 }, that is the smallest 𝜎(X, Y)-closed and convex set containing both A and 0.
  • Similarly, the bidual cone of a cone A is the 𝜎(X, Y)-closed conic hull of A.[7]
• If ℬ is a base at the origin for a TVS X, then X ' = B°.[8]
• If X is a locally convex TVS then the polars (taken with respect to ${\displaystyle \left\langle X,X^{\prime }\right\rangle }$) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of X' (i.e. given any bounded subset H of ${\displaystyle X_{\sigma }^{\prime }}$, there exists a neighborhood S of 0 in X such that H ⊆ S°).[6]
  • Conversely, if X is a locally convex TVS then the polars (taken with respect to ⟨X, X^#⟩) of any fundamental family of equicontinuous subsets of X' form a neighborhood base of the origin in X.[6]
• Let X be a TVS with a topology 𝜏. Then 𝜏 is a locally convex TVS topology if and only if 𝜏 is the topology of uniform convergence on the equicontinuous subsets of X'.[6] The last two results explain why equicontinuous subsets of the continuous dual space play such a prominent role in the modern theory of functional analysis: because equicontinuous subsets encapsulate all information about the locally convex space X's original topology.

Set relations

• ∅° = ∅^|r| = ∅^r = X and X° = X^|r| = X^r = { 0  }.[6]
• For all scalars s ≠ 0, (sA)° = 1/s (A°) and for all real t ≠ 0,  (tA)^|r| = 1/t (A^|r|) and (tA)^r = 1/t (A^r).
• A°°° = A°. However, for the real polar we have A^{r r r} ⊆ A^r.[6]
• For any finite collection of sets A₁, ..., A_n,

  (A₁) ∩ ··· ∩ A_n)°  =  (A₁°) ∪ ··· ∪ (A_n°).

• If A ⊆ B then B° ⊆ A°, B^r ⊆ B^r, and B^|r| ⊆ B^|r|.
  • An immediate corollary is that ${\displaystyle \bigcup _{i\in I}\left(A_{i}^{\circ }\right)\subseteq \left(\bigcap _{i\in I}A_{i}\right)^{\circ }}$; equality necessarily holds when I is finite and may fail to hold if I is infinite.
• ${\displaystyle \bigcap _{i\in I}\left(A_{i}^{\circ }\right)=\left(\bigcup _{i\in I}A_{i}\right)^{\circ }}$ and ${\displaystyle \bigcap _{i\in I}\left(A_{i}^{r}\right)=\left(\bigcup _{i\in I}A_{i}\right)^{r}}$.
• If C is a cone in X then C° = { y ∈ Y : ⟨c, y⟩ = 0 for all c ∈ C  }.[5]
• If ${\displaystyle \left(S_{i}\right)_{i\in I}}$ is a family of 𝜎(X, Y)-closed subsets of X containing 0 ∈ X, then the real polar of ${\displaystyle \cap _{i\in I}S_{i}}$ is the closed convex hull of ${\displaystyle \cup _{i\in I}\left(S_{i}^{r}\right)}$.[6]
• If 0 ∈ A ∩ B  then  A° ∩ B°  ⊆  2 [(A + B)°]  ⊆  2 (A° ∩ B°).[9]
• For a closed convex cone C in a real vector space X, the polar cone is the polar of C; that is,

  C° = { y ∈ Y : sup ⟨C, y⟩ ≤ 0 },

  that is, C° = { y ∈ Y : sup { ⟨c, y⟩ : c ∈ C } ≤ 0}.[1]

See also

• Bipolar theorem – theorem in convex analysis
• Polar cone
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Topological vector space – Vector space with a notion of nearness

Notes

1. Since for all of these completing definitions of the polar set A° to agree, if ⟨•, •⟩ is real-valued then it suffices for A to be symmetric, while if ⟨•, •⟩ is complex-valued then it suffices that ${\displaystyle e^{ir}}$A ⊆ A for all real s.

References

1. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
2. Narici & Beckenstein 2011, pp. 225-273.
3. Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. pp. 7–8. ISBN 978-9812380678.
4. Rockafellar, T.R. (1970). Convex Analysis. Princeton University. pp. 121-8. ISBN 978-0-691-01586-6.
5. Trèves 2006, pp. 195-201.
6. Schaefer & Wolff 1999, pp. 123–128.
7. Niculescu, C.P.; Persson, Lars-Erik (2018). Convex Functions and Their Applications. CMS Books in Mathematics. Cham, Switzerland: Springer. pp. 94–5, 134–5. doi:10.1007/978-3-319-78337-6. ISBN 978-3-319-78337-6.
8. Narici & Beckenstein 2011, p. 472.
9. Jarchow 1981, pp. 148-150.

Bibliography

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• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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