View source for Polar set

{{short description|Subset of all points that is bounded by some given point of a dual (in a dual pairing)}}
{{See also|Polar set (potential theory)}}

In [[Functional analysis|functional]] and [[convex analysis]], and related disciplines of [[mathematics]], the '''polar set''' <math>A^{\circ}</math> is a special convex set associated to any subset <math>A</math> of a [[vector space]] <math>X,</math> lying in the [[dual space]] <math>X^{\prime}.</math> 
The '''bipolar''' of a subset is the polar of <math>A^\circ,</math> but lies in <math>X</math> (not <math>X^{\prime\prime}</math>).

==Definitions==

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.<ref name="FctAnal" />{{citation needed|date=September 2018}} 
In each case, the definition describes a duality between certain subsets of a [[Dual system|pairing of vector spaces]] <math>\langle X, Y \rangle</math> over the real or complex numbers (<math>X</math> and <math>Y</math> are often [[topological vector space]]s (TVSs)).

If <math>X</math> is a vector space over the field <math>\mathbb{K}</math> then unless indicated otherwise, <math>Y</math> will usually, but not always, be some vector space of [[linear functional]]s on <math>X</math> and the dual pairing <math>\langle \cdot, \cdot \rangle : X \times Y \to \mathbb{K}</math> will be the [[Bilinear map|bilinear]] {{em|evaluation}} ({{em|at a point}}) {{em|map}} defined by 
<math display=block>\langle x, f \rangle := f(x).</math>
If <math>X</math> is a [[topological vector space]] then the space <math>Y</math> will usually, but not always, be the [[continuous dual space]] of <math>X,</math> in which case the dual pairing will again be the evaluation map.

Denote the closed ball of radius <math>r \geq 0</math> centered at the origin in the underlying scalar field <math>\mathbb{K}</math> of <math>X</math> by
<math display=block>B_r := B_r^{\mathbb{K}} := \{s \in \mathbb{K} : |s| \leq r\}.</math>

===Functional analytic definition===

====Absolute polar====

Suppose that <math>\langle X, Y \rangle</math> is a [[Dual system|pairing]]. 
The '''polar''' or '''absolute polar''' of a subset <math>A</math> of <math>X</math> is the set: 
<math display=block>\begin{alignat}{4}
A^{\circ} 
:=& \left\{y \in Y ~:~ \sup_{a \in A} |\langle a, y \rangle| \leq 1\right\} ~~~~&& \\[0.7ex]
 =& \left\{y \in Y ~:~ \sup |\langle A, y \rangle| \leq 1\right\} ~~~~&& \text{ where } |\langle A, y \rangle| := \{|\langle a, y \rangle| : a \in A\} \\[0.7ex]
 =& \left\{y \in Y ~:~ \langle A, y \rangle \subseteq B_1\right\} ~~~~&& \text{ where } B_1 := \{s \in \mathbb{K} : |s| \leq 1\}.\\[0.7ex]
\end{alignat}</math>

where <math>\langle A, y \rangle := \{\langle a, y \rangle : a \in A\}</math> denotes the [[Image (mathematics)|image]] of the set <math>A</math> under the map <math>\langle \cdot, y \rangle : X \to \mathbb{K}</math> defined by <math>x \mapsto \langle x, y \rangle.</math> 
If <math>\operatorname{cobal} A</math> denotes the [[convex balanced hull]] of <math>A,</math> which by definition is the smallest [[Convex set|convex]] and [[Balanced set|balanced]] subset of <math>X</math> that contains <math>A,</math> then <math>A^{\circ} = [\operatorname{cobal} A]^\circ.</math>

This is an [[Affine transformation|affine shift]] of the geometric definition; 
it has the useful characterization that the functional-analytic polar of the unit ball (in <math>X</math>) is precisely the unit ball (in <math>Y</math>).

The '''prepolar''' or '''absolute prepolar''' of a subset <math>B</math> of <math>Y</math> is the set:
<math display=block>{}^{\circ} B 
:= \left\{x \in X ~:~ \sup_{b \in B} |\langle x, b \rangle| \leq 1\right\} 
 = \{x \in X ~:~ \sup |\langle x, B \rangle| \leq 1\}</math>

Very often, the prepolar of a subset <math>B</math> of <math>Y</math> is also called the '''polar''' or '''absolute polar''' of <math>B</math> and denoted by <math>B^{\circ}</math>; 
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 <math>A</math> of <math>X,</math> often denoted by <math>A^{\circ \circ},</math> is the set <math>{}^{\circ}\left(A^{\circ}\right)</math>; 
that is, 
<math display=block>A^{\circ \circ} := {}^{\circ}\left(A^{\circ}\right) 
= \left\{x \in X ~:~ \sup_{y \in A^{\circ}} |\langle x, y \rangle| \leq 1\right\}.</math>

====Real polar====

The '''real polar''' of a subset <math>A</math> of <math>X</math> is the set: 
<math display=block>A^r := \left\{y \in Y ~:~ \sup_{a \in A} \operatorname{Re} \langle a, y \rangle \leq 1\right\}</math>
and the '''real prepolar''' of a subset <math>B</math> of <math>Y</math> is the set:
<math display=block>{}^r B := \left\{x \in X ~:~ \sup_{b \in B} \operatorname{Re} \langle x, b \rangle \leq 1\right\}.</math>

As with the absolute prepolar, the real prepolar is usually called the '''real polar''' and is also denoted by <math>B^r.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
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 <math>A^{\circ}</math> for it (rather than the notation <math>A^r</math> that is used in this article and in [Narici 2011]).

The '''real bipolar''' of a subset <math>A</math> of <math>X,</math> sometimes denoted by <math>A^{r r},</math> is the set <math>{}^{r}\left(A^r\right)</math>; 
it is equal to the <math>\sigma(X, Y)</math>-closure of the [[convex hull]] of <math>A \cup \{0\}.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}

For a subset <math>A</math> of <math>X,</math> <math>A^r</math>  is convex, <math>\sigma(Y, X)</math>-closed, and contains <math>A^{\circ}.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
In general, it is possible that <math>A^{\circ} \neq A^r</math> but equality will hold if <math>A</math> is [[Balanced set|balanced]]. 
Furthermore, <math>A^{\circ} = \left(\operatorname{bal} \left(A^{r}\right)\right)</math> where <math>\operatorname{bal} \left(A^{r}\right)</math> denotes the [[Balanced set|balanced hull]] of <math>A^r.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}

====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 <math>A^{\circ}</math> almost always represents {{em|their}} choice of the definition (so the meaning of the notation <math>A^{\circ}</math> may vary from source to source). 
In particular, the polar of <math>A</math> is sometimes defined as:
<math display=block>A^{|r|} := \left\{y \in Y ~:~ \sup_{a \in A} |\operatorname{Re} \langle a, y \rangle| \leq 1\right\}</math>
where the notation <math>A^{|r|}</math> is {{em|not}} standard notation.

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

It is always the case that
<math display=block>A^{\circ} ~ \subseteq ~ A^{|r|} ~ \subseteq ~ A^r</math>
and if <math>\langle \cdot, \cdot \rangle</math> is real-valued (or equivalently, if <math>X</math> and <math>Y</math> are vector spaces over <math>\R</math>) then <math>A^{\circ} = A^{|r|}.</math>

If <math>A</math> is a [[symmetric set]] (that is, <math>-A = A</math> or equivalently, <math>-A \subseteq A</math>) then <math>A^{|r|} = A^r</math> where if in addition <math>\langle \cdot, \cdot \rangle</math> is real-valued then <math>A^{\circ} = A^{|r|} = A^r.</math>

If <math>X</math> and <math>Y</math> are vector spaces over <math>\C</math> (so that <math>\langle \cdot, \cdot \rangle</math> is complex-valued) and if <math>i A \subseteq A</math> (where note that this implies <math>-A = A</math> and <math>i A = A</math>), then
<math display=block>A^{\circ} \subseteq A^{|r|} = A^r \subseteq \left(\tfrac{1}{\sqrt{2}} A\right)^{\circ}</math>
where if in addition <math>e^{i r} A \subseteq A</math> for all real <math>r</math> then <math>A^{\circ} = A^r.</math>

Thus for all of these definitions of the polar set of <math>A</math> to agree, it suffices that <math>s A \subseteq A</math> for all scalars <math>s</math> of [[unit length]]<ref group=note>
Since for all of these completing definitions of the polar set <math>A^{\circ}</math> to agree, if <math>\langle \cdot, \cdot \rangle</math> is real-valued then it suffices for <math>A</math> to be symmetric, while if <math>\langle \cdot, \cdot \rangle</math> is complex-valued then it suffices that <math>e^{i r} A \subseteq A</math> for all real <math>s.</math>
</ref> (where this is equivalent to <math>s A = A</math> for all unit length scalar <math>s</math>).
In particular, all definitions of the polar of <math>A</math> agree when <math>A</math> 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 differences in the definitions of the "polar" of a set <math>A</math> do sometimes introduce subtle or important technical differences when <math>A</math> is not necessarily balanced.

====Specialization for the canonical duality====

'''Algebraic dual space'''

If <math>X</math> is any vector space then let <math>X^{\#}</math> denote the [[algebraic dual space]] of <math>X,</math> which is the set of all [[linear functionals]] on <math>X.</math> The vector space <math>X^{\#}</math> is always a closed subset of the space <math>\mathbb{K}^X</math> of all <math>\mathbb{K}</math>-valued functions on <math>X</math> under the topology of pointwise convergence so when <math>X^{\#}</math> is endowed with the subspace topology, then <math>X^{\#}</math> becomes a [[Hausdorff space|Hausdorff]] [[Complete topological vector space|complete]] [[Locally convex topological vector space|locally convex]] [[topological vector space]] (TVS). 
For any subset <math>A \subseteq X,</math> let
<math display=block>\begin{alignat}{4}
A^{\#} := A^{\circ, \#} 
:=& \left\{f \in X^{\#} ~:~ \sup_{a \in A} |f(a)| \leq 1\right\} && \\[0.7ex]
 =& \left\{f \in X^{\#}  ~:~ \sup |f(A)| \leq 1\right\}  ~~~~&& \text{ where } |f(A)| := \{|f(a)| : a \in A\} \\[0.7ex]
 =& \left\{f \in X^{\#}  ~:~ f(A) \subseteq B_1\right\}  ~~~&& \text{ where } B_1 := \{s \in \mathbb{K} : |s| \leq 1\}.\\[0.7ex]
\end{alignat}</math>

If <math>A \subseteq B \subseteq X</math> are any subsets then <math>B^{\#} \subseteq A^{\#}</math> and <math>A^{\#} = [\operatorname{cobal} A]^{\#},</math> where <math>\operatorname{cobal} A</math> denotes the [[convex balanced hull]] of <math>A.</math> 
For any finite-dimensional vector subspace <math>Y</math> of <math>X,</math> let <math>\tau_Y</math> denote the [[Euclidean topology]] on <math>Y,</math> which is the unique topology that makes <math>Y</math> into a [[Hausdorff space|Hausdorff]] topological vector space (TVS). 
If <math>A_{\cup \operatorname{cl} \operatorname{Finite}}</math> denotes the union of all [[Closure (topology)|closures]] <math>\operatorname{cl}_{\left(Y, \tau_Y\right)} (Y \cap A)</math> as <math>Y</math> varies over all finite dimensional vector subspaces of <math>X,</math> then <math>A^{\#} = \left[A_{\cup \operatorname{cl} \operatorname{Finite}}\right]^{\#}</math> (see this footnote<ref group=note>To prove that <math>A^{\#} \subseteq \left[A_{\cup \operatorname{cl} \operatorname{Finite}}\right]^{\#},</math> let <math>f \in A^{\#}.</math> If <math>Y</math> is a finite-dimensional vector subspace of <math>X</math> then because <math>f\big\vert_Y : \left(Y, \tau_Y\right) \to \mathbb{K}</math> is continuous (as is true of all linear functionals on a finite-dimensional Hausdorff TVS), it follows from <math>f(A) \subseteq B_1</math> and <math>B_1</math> being a closed set that <math>f\left(\operatorname{cl}_{\left(Y, \tau_Y\right)} (Y \cap A)\right) = f\big\vert_Y \left(\operatorname{cl}_{\left(Y, \tau_Y\right)} (Y \cap A)\right) \subseteq \operatorname{cl}_{\mathbb{K}}(f(Y \cap A)) \subseteq \operatorname{cl}_{\mathbb{K}} f(A) \subseteq \operatorname{cl}_{\mathbb{K}} B_1 = B_1.</math> The union of all such sets is consequently also a subset of <math>B_1,</math> which proves that <math>f\left(A_{\cup \operatorname{cl} \operatorname{Finite}}\right) \subseteq B_1</math> and so <math>f \in \left[A_{\cup \operatorname{cl} \operatorname{Finite}}\right]^{\#}.</math> <math>\blacksquare</math> In general, if <math>\tau</math> is any TVS-topology on <math>X</math> then <math>A_{\cup \operatorname{cl} \operatorname{Finite}} \subseteq \operatorname{cl}_{(X, \tau)} A.</math></ref> 
for an explanation). 
If <math>A</math> is an absorbing subset of <math>X</math> then by the [[Banach–Alaoglu theorem]], <math>A^{\#}</math> is a [[Weak-* topology|weak-* compact]] subset of <math>X^{\#}.</math>

If <math>A \subseteq X</math> is any non-empty subset of a vector space <math>X</math> and if <math>Y</math> is any vector space of linear functionals on <math>X</math> (that is, a vector subspace of the [[algebraic dual space]] of <math>X</math>) then the real-valued map

:<math>|\,\cdot\,|_A \;:\, Y \,\to\, \Reals</math> {{space|4}} defined by {{space|4}} <math>\left|x^{\prime}\right|_A ~:=~ \sup \left|x^{\prime}(A)\right| ~:=~ \sup_{a \in A} \left|x^{\prime}(a)\right|</math>

is a [[seminorm]] on <math>Y.</math> If <math>A = \varnothing</math> then by definition of the [[supremum]], <math>\, \sup \left| x^{\prime}(A) \right| = -\infty \,</math> so that the map <math>\, |\,\cdot\,|_{\varnothing} = -\infty \,</math> defined above would not be real-valued and consequently, it would not be a seminorm.

'''Continuous dual space'''

Suppose that <math>X</math> is a [[topological vector space]] (TVS) with [[continuous dual space]] <math>X^{\prime}.</math> 
The important special case where <math>Y := X^{\prime}</math> and the brackets represent the canonical map:
<math display=block>\left\langle x, x^{\prime} \right\rangle := x^{\prime}(x)</math>
is now considered. 
The triple <math>\left\langle X, X^{\prime} \right\rangle</math> is the called the {{em|canonical [[Dual system|pairing]]}} associated with <math>X.</math>

The polar of a subset <math>A \subseteq X</math> with respect to this canonical pairing is:
<math display=block>\begin{alignat}{4}
A^{\circ} 
:=& \left\{x^{\prime} \in X^{\prime} ~:~ \sup_{a \in A} \left|x^{\prime}(a)\right| \leq 1\right\} ~~~~&& \text{ because } \left\langle a, x^{\prime} \right\rangle := x^{\prime}(a)  \\[0.7ex]
 =& \left\{x^{\prime} \in X^{\prime} ~:~ \sup \left|x^{\prime}(A)\right| \leq 1\right\} ~~~~&& \text{ where } \left|x^{\prime}(A)\right| := \left\{\left|x^{\prime}(a)\right| : a \in A\right\} \\[0.7ex]
 =& \left\{x^{\prime} \in X^{\prime} ~:~ x^{\prime}(A) \subseteq B_1\right\} ~~~~&&  \text{ where } B_1 := \{s \in \mathbb{K} : |s| \leq 1\}.\\[0.7ex]
\end{alignat}</math>

For any subset <math>A \subseteq X,</math> <math>A^{\circ} = \left[\operatorname{cl}_X A\right]^{\circ}</math> where <math>\operatorname{cl}_X A</math> denotes the [[Closure (topology)|closure]] of <math>A</math> in <math>X.</math>

The [[Banach–Alaoglu theorem]] states that if <math>A \subseteq X</math> is a neighborhood of the origin in <math>X</math> then <math>A^{\circ} = A^{\#}</math> and this polar set is a [[Compact space|compact subset]] of the continuous dual space <math>X^{\prime}</math> when <math>X^{\prime}</math> is endowed with the [[weak-* topology]] (also known as the topology of pointwise convergence).

If <math>A</math> satisfies <math>s A \subseteq A</math> for all scalars <math>s</math> of unit length then one may replace the absolute value signs by <math>\operatorname{Re}</math> (the real part operator) so that:
<math display=block>\begin{alignat}{4}
A^{\circ} = A^r 
:=& \left\{x^{\prime} \in X^{\prime} ~:~ \sup_{a \in A} \operatorname{Re} x^{\prime}(a) \leq 1\right\} \\[0.7ex]
 =& \left\{x^{\prime} \in X^{\prime} ~:~ \sup \operatorname{Re} x^{\prime}(A) \leq 1\right\}. \\[0.7ex]
\end{alignat}</math>

The prepolar of a subset <math>B</math> of <math>Y = X^{\prime}</math> is:
<math display=block>{}^{\circ} B := \left\{x \in X ~:~ \sup_{b^{\prime} \in B} \left|b^{\prime}(x)\right| \leq 1\right\} 
= \{x \in X : \sup |B(x)| \leq 1\}</math>

If <math>B</math> satisfies <math>s B \subseteq B</math> for all scalars <math>s</math> of unit length then one may replace the absolute value signs with <math>\operatorname{Re}</math> so that:
<math display=block>{}^{\circ} B = \left\{x \in X ~:~ \sup_{b^{\prime} \in B} \operatorname{Re} b^{\prime}(x) \leq 1\right\} 
= \{x \in X ~:~ \sup \operatorname{Re} B(x) \leq 1\}</math>
where <math>B(x) := \left\{b^{\prime}(x) ~:~ b^{\prime} \in B\right\}.</math>

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

If <math>X</math> is a normed space and <math>S</math> is the open or closed unit ball in <math>X</math> (or even any subset of the closed unit ball that contains the open unit ball) then <math>S^{\circ}</math> is the closed unit ball in the continuous dual space <math>X^{\prime}</math> when <math>X^{\prime}</math> is endowed with its canonical [[dual norm]].

===Geometric definition for cones===
{{main|Pole and polar}}

The [[Dual cone|polar cone]] of a convex cone <math>A \subseteq X</math> is the set
<math display=block>A^{\circ} := \left\{y \in Y ~:~ \sup_{x \in A} \langle x, y \rangle \leq 0\right\}</math>

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 <math>x \in X</math> is the locus <math>\{y ~:~ \langle y, x \rangle = 0\}</math>; 
the [[Projective dual|dual]] relationship for a hyperplane yields that hyperplane's polar point.<ref name="VecBook">{{cite book|title=Convex Analysis in General Vector Spaces|url=https://archive.org/details/convexanalysisge00zali_934|url-access=limited|last=Zălinescu|first=C.|publisher=World Scientific|year=2002|isbn=978-9812380678|location=River Edge, NJ|pages=[https://archive.org/details/convexanalysisge00zali_934/page/n27 7]–8}}</ref>{{citation needed|date=September 2018}}

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.<ref name="ConvAnal">{{cite book|title=Convex Analysis|url=https://archive.org/details/convexanalysis00rock|url-access=limited|last=Rockafellar|first=T.R.|pages=[https://archive.org/details/convexanalysis00rock/page/n137 121]-8|publisher=Princeton University|year=1970|isbn=978-0-691-01586-6}}</ref>

==Properties==

Unless stated otherwise, <math>\langle X, Y \rangle</math> will be a [[Dual system|pairing]]. 
The topology <math>\sigma(Y, X)</math> is the [[weak-* topology]] on <math>Y</math> while <math>\sigma(X, Y)</math> is the [[weak topology]] on <math>X.</math> 
For any set <math>A,</math> <math>A^r</math> denotes the real polar of <math>A</math> and <math>A^{\circ}</math> denotes the absolute polar of <math>A.</math> 
The term "polar" will refer to the {{em|absolute}} polar.

* The (absolute) polar of a set is [[Convex set|convex]] and [[Balanced set|balanced]].{{sfn|Trèves|2006|pp=195-201}}
* The real polar <math>A^r</math> of a subset <math>A</math> of <math>X</math> is convex but {{em|not}} necessarily balanced; <math>A^r</math> will be balanced if <math>A</math> is balanced.{{sfn|Schaefer|Wolff|1999|pp=123–128}}
* If <math>s A \subseteq A</math> for all scalars <math>s</math> of unit length then <math>A^{\circ} = A^r.</math>
* <math>A^{\circ}</math> is [[Closed set|closed]] in <math>Y</math> under the [[Weak-star operator topology|weak-*-topology]] on <math>Y</math>.<ref name="VecBook" />
* A subset <math>S</math> of <math>X</math> is weakly bounded (i.e. <math>\sigma(X, Y)</math>-bounded) if and only if <math>S^{\circ}</math> is [[absorbing set|absorbing]] in <math>Y</math>.{{sfn|Narici|Beckenstein|2011|pp=225-273}}
* For a dual pair <math>\langle X, X^{\prime} \rangle,</math> where <math>X</math> is a TVS and <math>X^{\prime}</math> is its continuous dual space, if <math>B \subseteq X</math> is bounded then <math>B^{\circ}</math> is [[Absorbing set|absorbing]] in <math>X^{\prime}.</math>{{sfn|Trèves|2006|pp=195-201}} If <math>X</math> is locally convex and <math>B^{\circ}</math> is absorbing in <math>X^{\prime}</math> then <math>B</math> is bounded in <math>X.</math> Moreover, a subset <math>S</math> of <math>X</math> is weakly bounded if and only if <math>S^{\circ}</math> is [[Absorbing set|absorbing]] in <math>X^{\prime}.</math>
* The bipolar <math>A^{\circ\circ}</math> of a set <math>A</math> is the <math>\sigma(X, Y)</math>-[[closed convex hull]] of <math>A \cup \{0\},</math> that is the smallest <math>\sigma(X, Y)</math>-closed and convex set containing both <math>A</math> and <math>0.</math> 
** Similarly, the bidual cone of a cone <math>A</math> is the <math>\sigma(X, Y)</math>-closed [[conic hull]] of <math>A.</math><ref name="ConvFct">{{cite book|pages=94–5,134–5|title=Convex Functions and Their Applications|last1=Niculescu|first1=C.P.|last2=Persson|first2=Lars-Erik|series=CMS Books in Mathematics|publisher=Springer|isbn=978-3-319-78337-6|location=Cham, Switzerland|doi=10.1007/978-3-319-78337-6|year=2018}}</ref>
* If <math>\mathcal{B}</math> is a base at the origin for a TVS <math>X</math> then <math>X^{\prime} = \bigcup_{B \in \mathbb{B}} \left(B^{\circ}\right).</math>{{sfn|Narici|Beckenstein|2011|p=472}}
* If <math>X</math> is a locally convex TVS then the polars (taken with respect to <math>\left\langle X, X^{\prime} \right\rangle</math>) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of <math>X^{\prime}</math> (i.e. given any bounded subset <math>H</math> of <math>X^{\prime}_{\sigma},</math> there exists a neighborhood <math>S</math> of the origin in <math>X</math> such that <math>H \subseteq S^{\circ}</math>).{{sfn|Schaefer|Wolff|1999|pp=123–128}} 
** Conversely, if <math>X</math> is a locally convex TVS then the polars (taken with respect to <math>\langle X, X^{\#} \rangle</math>) of any fundamental family of equicontinuous subsets of <math>X^{\prime}</math> form a neighborhood base of the origin in <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=123–128}}
* Let <math>X</math> be a TVS with a topology <math>\tau.</math> Then <math>\tau</math> is a locally convex TVS topology if and only if <math>\tau</math> is the topology of uniform convergence on the equicontinuous subsets of <math>X^{\prime}.</math>{{sfn|Schaefer|Wolff|1999|pp=123–128}} 
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 <math>X</math>'s original topology.

'''Set relations'''

* <math>X^{\circ} = X^{|r|} = X^r = \{0\}</math>{{sfn|Schaefer|Wolff|1999|pp=123–128}} and <math>\varnothing^{\circ} = \varnothing^{|r|} = \varnothing^r = Y.</math>
* For all scalars <math>s \neq 0,</math> <math>(s A)^{\circ} = \tfrac{1}{s} \left(A^{\circ}\right)</math> and for all real <math>t \neq 0,</math> <math>(t A)^{|r|} = \tfrac{1}{t} \left(A^{|r|}\right)</math> and <math>(t A)^r = \tfrac{1}{t} \left(A^r\right).</math>
* <math>A^{\circ\circ\circ} = A^{\circ}.</math> However, for the real polar we have <math>A^{r r r} \subseteq A^r.</math>{{sfn|Schaefer|Wolff|1999|pp=123–128}}
* For any finite collection of sets <math>A_1, \ldots, A_n,</math> <math display=block>\left(A_1 \cap \cdots \cap A_n\right)^{\circ} = \left(A_1^{\circ}\right) \cup \cdots \cup \left(A_n^{\circ}\right).</math>
* If <math>A \subseteq  B</math> then <math>B^{\circ} \subseteq  A^{\circ},</math> <math>B^r \subseteq  A^r,</math> and <math>B^{|r|} \subseteq  A^{|r|}.</math>
** An immediate corollary is that <math>\bigcup_{i \in I} \left(A_i^{\circ}\right) \subseteq \left(\bigcap_{i \in I} A_i\right)^{\circ}</math>; equality necessarily holds when <math>I</math> is finite and may fail to hold if <math>I</math> is infinite.
* <math>\bigcap_{i \in I} \left(A_i^{\circ}\right) = \left(\bigcup_{i \in I} A_i\right)^{\circ}</math> and <math>\bigcap_{i \in I} \left(A_i^r\right) = \left(\bigcup_{i \in I} A_i\right)^r.</math>
* If <math>C</math> is a cone in <math>X</math> then <math>C^{\circ} = \left\{ y \in Y : \langle c, y \rangle = 0 \text{ for all } c \in C \right\}.</math>{{sfn|Trèves|2006|pp=195-201}}
* If <math>\left(S_i\right)_{i \in I}</math> is a family of <math>\sigma(X, Y)</math>-closed subsets of <math>X</math> containing <math>0 \in X,</math> then the real polar of <math>\cap_{i \in I} S_i</math> is the closed convex hull of <math>\cup_{i \in I} \left(S_i^r\right).</math>{{sfn|Schaefer|Wolff|1999|pp=123–128}}
* If <math>0 \in A \cap B</math> then <math>A^{\circ} \cap B^{\circ} \subseteq 2 \left[(A + B)^{\circ}\right] \subseteq 2\left(A^{\circ} \cap B^{\circ}\right).</math>{{sfn|Jarchow|1981|pp=148-150}}
* For a closed [[convex cone]] <math>C</math> in a real vector space <math>X,</math> the [[polar cone]] is the polar of <math>C</math>; that is, <math display=block>C^{\circ} = \{y \in Y : \sup_{} \langle C, y \rangle \leq 0\},</math> where <math>\sup_{} \langle C, y \rangle := \sup_{c \in C} \langle c, y \rangle.</math><ref name="FctAnal">{{cite book|last1=Aliprantis|first1=C.D.|last2=Border|first2=K.C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=215}}</ref>

==See also==

* {{annotated link|Banach–Alaoglu theorem}}
* {{annotated link|Bipolar theorem}}
* {{annotated link|Polar cone}}
* {{annotated link|Polar topology}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Topological vector space}}

==Notes==

{{reflist|group=note}}

==References==

{{reflist}}

==Bibliography==

* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|1980|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Treves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Duality and spaces of linear maps}}
{{Topological vector spaces}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Linear functionals]]
[[Category:Topological vector spaces]]