Polar set

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.{{citation needed|date=September 2018}}

In each case, the definition describes a duality between certain subsets of a pairing of vector spaces $\backslash langle\; X,\; Y\; \backslash rangle$ over the real or complex numbers ($X$ and $Y$ are often topological vector spaces (TVSs)).

If $X$ is a vector space over the field $\backslash mathbb\{K\}$ then unless indicated otherwise, $Y$ will usually, but not always, be some vector space of linear functionals on $X$ and the dual pairing $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; :\; X\; \backslash times\; Y\; \backslash to\; \backslash mathbb\{K\}$ will be the bilinear {{em|evaluation}} ({{em|at a point}}) {{em|map}} defined by

$$\backslash langle\; x,\; f\; \backslash rangle\; :=\; f(x).$$

If $X$ is a topological vector space then the space $Y$ will usually, but not always, be the continuous dual space of $X,$ in which case the dual pairing will again be the evaluation map.

Denote the closed ball of radius $r\; \backslash geq\; 0$ centered at the origin in the underlying scalar field $\backslash mathbb\{K\}$ of $X$ by

$$B\_r\; :=\; B\_r^\{\backslash mathbb\{K\}\}\; :=\; \backslash \{s\; \backslash in\; \backslash mathbb\{K\}\; :\; |s|\; \backslash leq\; r\backslash \}.$$

Suppose that $\backslash langle\; X,\; Y\; \backslash rangle$ is a pairing.

The polar or absolute polar of a subset $A$ of $X$ is the set:

$$\backslash begin\{alignat\}\{4\}$$

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}

where $\backslash langle\; A,\; y\; \backslash rangle\; :=\; \backslash \{\backslash langle\; a,\; y\; \backslash rangle\; :\; a\; \backslash in\; A\backslash \}$ denotes the image of the set $A$ under the map $\backslash langle\; \backslash cdot,\; y\; \backslash rangle\; :\; X\; \backslash to\; \backslash mathbb\{K\}$ defined by $x\; \backslash mapsto\; \backslash langle\; x,\; y\; \backslash rangle.$

If $\backslash operatorname\{cobal\}\; A$ denotes the convex balanced hull of $A,$ which by definition is the smallest convex and balanced subset of $X$ that contains $A,$ then $A^\{\backslash circ\}\; =\; [\backslash operatorname\{cobal\}\; A]^\backslash circ.$

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:

$$\{\}^\{\backslash circ\}\; B$$

:= \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\}

Very often, the prepolar of a subset $B$ of $Y$ is also called the polar or absolute polar of $B$ and denoted by $B^\{\backslash circ\}$;

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 $A^\{\backslash circ\; \backslash circ\},$ is the set $\{\}^\{\backslash circ\}\backslash left(A^\{\backslash circ\}\backslash right)$;

that is,

$$A^\{\backslash circ\; \backslash circ\}\; :=\; \{\}^\{\backslash circ\}\backslash left(A^\{\backslash circ\}\backslash right)$$

= \left\{x \in X ~:~ \sup_{y \in A^{\circ}} |\langle x, y \rangle| \leq 1\right\}.

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

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

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

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

As with the absolute prepolar, the real prepolar is usually called the real polar and is also denoted by $B^r.${{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 $A^\{\backslash circ\}$ 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 $\{\}^\{r\}\backslash left(A^r\backslash right)$;

it is equal to the $\backslash sigma(X,\; Y)$-closure of the convex hull of $A\; \backslash cup\; \backslash \{0\backslash \}.${{sfn|Narici|Beckenstein|2011|pp=225-273}}

For a subset $A$ of $X,$ $A^r$ is convex, $\backslash sigma(Y,\; X)$-closed, and contains $A^\{\backslash circ\}.${{sfn|Narici|Beckenstein|2011|pp=225-273}}

In general, it is possible that $A^\{\backslash circ\}\; \backslash neq\; A^r$ but equality will hold if $A$ is balanced.

Furthermore, $A^\{\backslash circ\}\; =\; \backslash left(\backslash operatorname\{bal\}\; \backslash left(A^\{r\}\backslash right)\backslash right)$ where $\backslash operatorname\{bal\}\; \backslash left(A^\{r\}\backslash right)$ denotes the balanced hull of $A^r.${{sfn|Narici|Beckenstein|2011|pp=225-273}}

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 $A^\{\backslash circ\}$ almost always represents {{em|their}} choice of the definition (so the meaning of the notation $A^\{\backslash circ\}$ may vary from source to source).

In particular, the polar of $A$ is sometimes defined as:

$$A^$$

where the notation $A^$

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

It is always the case that

$$A^\{\backslash circ\}\; ~\; \backslash subseteq\; ~\; A^$$

and if $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is real-valued (or equivalently, if $X$ and $Y$ are vector spaces over $\backslash R$) then $A^\{\backslash circ\}\; =\; A^$

If $A$ is a symmetric set (that is, $-A\; =\; A$ or equivalently, $-A\; \backslash subseteq\; A$) then $A^$

If $X$ and $Y$ are vector spaces over $\backslash C$ (so that $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is complex-valued) and if $i\; A\; \backslash subseteq\; A$ (where note that this implies $-A\; =\; A$ and $i\; A\; =\; A$), then

$$A^\{\backslash circ\}\; \backslash subseteq\; A^$$

where if in addition $e^\{i\; r\}\; A\; \backslash subseteq\; A$ for all real $r$ then $A^\{\backslash circ\}\; =\; A^r.$

Thus for all of these definitions of the polar set of $A$ to agree, it suffices that $s\; A\; \backslash subseteq\; A$ for all scalars $s$ of unit length

Since for all of these completing definitions of the polar set $A^\{\backslash circ\}$ to agree, if $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is real-valued then it suffices for $A$ to be symmetric, while if $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is complex-valued then it suffices that $e^\{i\; r\}\; A\; \backslash subseteq\; A$ for all real $s.$

(where this is equivalent to $s\; A\; =\; A$ 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 differences in the definitions of the "polar" of a set $A$ do sometimes introduce subtle or important technical differences when $A$ is not necessarily balanced.

Algebraic dual space

If $X$ is any vector space then let $X^\{\backslash \#\}$ denote the algebraic dual space of $X,$ which is the set of all linear functionals on $X.$ The vector space $X^\{\backslash \#\}$ is always a closed subset of the space $\backslash mathbb\{K\}^X$ of all $\backslash mathbb\{K\}$-valued functions on $X$ under the topology of pointwise convergence so when $X^\{\backslash \#\}$ is endowed with the subspace topology, then $X^\{\backslash \#\}$ becomes a Hausdorff complete locally convex topological vector space (TVS).

For any subset $A\; \backslash subseteq\; X,$ let

$$\backslash begin\{alignat\}\{4\}$$

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}

If $A\; \backslash subseteq\; B\; \backslash subseteq\; X$ are any subsets then $B^\{\backslash \#\}\; \backslash subseteq\; A^\{\backslash \#\}$ and $A^\{\backslash \#\}\; =\; [\backslash operatorname\{cobal\}\; A]^\{\backslash \#\},$ where $\backslash operatorname\{cobal\}\; A$ denotes the convex balanced hull of $A.$

For any finite-dimensional vector subspace $Y$ of $X,$ let $\backslash tau\_Y$ denote the Euclidean topology on $Y,$ which is the unique topology that makes $Y$ into a Hausdorff topological vector space (TVS).

If $A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}$ denotes the union of all closures $\backslash operatorname\{cl\}\_\{\backslash left(Y,\; \backslash tau\_Y\backslash right)\}\; (Y\; \backslash cap\; A)$ as $Y$ varies over all finite dimensional vector subspaces of $X,$ then $A^\{\backslash \#\}\; =\; \backslash left[A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}\backslash right]^\{\backslash \#\}$ (see this footnoteTo prove that $A^\{\backslash \#\}\; \backslash subseteq\; \backslash left[A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}\backslash right]^\{\backslash \#\},$ let $f\; \backslash in\; A^\{\backslash \#\}.$ If $Y$ is a finite-dimensional vector subspace of $X$ then because $f\backslash big\backslash vert\_Y\; :\; \backslash left(Y,\; \backslash tau\_Y\backslash right)\; \backslash to\; \backslash mathbb\{K\}$ is continuous (as is true of all linear functionals on a finite-dimensional Hausdorff TVS), it follows from $f(A)\; \backslash subseteq\; B\_1$ and $B\_1$ being a closed set that $f\backslash left(\backslash operatorname\{cl\}\_\{\backslash left(Y,\; \backslash tau\_Y\backslash right)\}\; (Y\; \backslash cap\; A)\backslash right)\; =\; f\backslash big\backslash vert\_Y\; \backslash left(\backslash operatorname\{cl\}\_\{\backslash left(Y,\; \backslash tau\_Y\backslash right)\}\; (Y\; \backslash cap\; A)\backslash right)\; \backslash subseteq\; \backslash operatorname\{cl\}\_\{\backslash mathbb\{K\}\}(f(Y\; \backslash cap\; A))\; \backslash subseteq\; \backslash operatorname\{cl\}\_\{\backslash mathbb\{K\}\}\; f(A)\; \backslash subseteq\; \backslash operatorname\{cl\}\_\{\backslash mathbb\{K\}\}\; B\_1\; =\; B\_1.$ The union of all such sets is consequently also a subset of $B\_1,$ which proves that $f\backslash left(A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}\backslash right)\; \backslash subseteq\; B\_1$ and so $f\; \backslash in\; \backslash left[A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}\backslash right]^\{\backslash \#\}.$ $\backslash blacksquare$ In general, if $\backslash tau$ is any TVS-topology on $X$ then $A\_\{\backslash cup\; \backslash operatorname\{cl\}\; \backslash operatorname\{Finite\}\}\; \backslash subseteq\; \backslash operatorname\{cl\}\_\{(X,\; \backslash tau)\}\; A.$

for an explanation).

If $A$ is an absorbing subset of $X$ then by the Banach–Alaoglu theorem, $A^\{\backslash \#\}$ is a weak-* compact subset of $X^\{\backslash \#\}.$

If $A\; \backslash subseteq\; X$ is any non-empty subset of a vector space $X$ and if $Y$ is any vector space of linear functionals on $X$ (that is, a vector subspace of the algebraic dual space of $X$) then the real-valued map

:$|\backslash ,\backslash cdot\backslash ,|\_A\; \backslash ;:\backslash ,\; Y\; \backslash ,\backslash to\backslash ,\; \backslash Reals$ {{space|4}} defined by {{space|4}} $\backslash left|x^\{\backslash prime\}\backslash right|\_A\; ~:=~\; \backslash sup\; \backslash left|x^\{\backslash prime\}(A)\backslash right|\; ~:=~\; \backslash sup\_\{a\; \backslash in\; A\}\; \backslash left|x^\{\backslash prime\}(a)\backslash right|$

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

Continuous dual space

Suppose that $X$ is a topological vector space (TVS) with continuous dual space $X^\{\backslash prime\}.$

The important special case where $Y\; :=\; X^\{\backslash prime\}$ and the brackets represent the canonical map:

$$\backslash left\backslash langle\; x,\; x^\{\backslash prime\}\; \backslash right\backslash rangle\; :=\; x^\{\backslash prime\}(x)$$

is now considered.

The triple $\backslash left\backslash langle\; X,\; X^\{\backslash prime\}\; \backslash right\backslash rangle$ is the called the {{em|canonical pairing}} associated with $X.$

The polar of a subset $A\; \backslash subseteq\; X$ with respect to this canonical pairing is:

$$\backslash begin\{alignat\}\{4\}$$

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}

For any subset $A\; \backslash subseteq\; X,$ $A^\{\backslash circ\}\; =\; \backslash left[\backslash operatorname\{cl\}\_X\; A\backslash right]^\{\backslash circ\}$ where $\backslash operatorname\{cl\}\_X\; A$ denotes the closure of $A$ in $X.$

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

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

$$\backslash begin\{alignat\}\{4\}$$

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}

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

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

= \{x \in X : \sup |B(x)| \leq 1\}

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

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

= \{x \in X ~:~ \sup \operatorname{Re} B(x) \leq 1\}

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

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 $S^\{\backslash circ\}$ is the closed unit ball in the continuous dual space $X^\{\backslash prime\}$ when $X^\{\backslash prime\}$ is endowed with its canonical dual norm.

{{main|Pole and polar}}

The polar cone of a convex cone $A\; \backslash subseteq\; X$ is the set

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

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 $x\; \backslash in\; X$ is the locus $\backslash \{y\; ~:~\; \backslash langle\; y,\; x\; \backslash rangle\; =\; 0\backslash \}$;

the dual relationship for a hyperplane yields that hyperplane's polar point.{{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}}{{citation needed|date=September 2018}}

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

Unless stated otherwise, $\backslash langle\; X,\; Y\; \backslash rangle$ will be a pairing.

The topology $\backslash sigma(Y,\; X)$ is the weak-* topology on $Y$ while $\backslash sigma(X,\; Y)$ is the weak topology on $X.$

For any set $A,$ $A^r$ denotes the real polar of $A$ and $A^\{\backslash circ\}$ denotes the absolute polar of $A.$

The term "polar" will refer to the {{em|absolute}} polar.

• The (absolute) polar of a set is convex and balanced.{{sfn|Trèves|2006|pp=195-201}}
• The real polar $A^r$ of a subset $A$ of $X$ is convex but {{em|not}} necessarily balanced; $A^r$ will be balanced if $A$ is balanced.{{sfn|Schaefer|Wolff|1999|pp=123–128}}
• If $s\; A\; \backslash subseteq\; A$ for all scalars $s$ of unit length then $A^\{\backslash circ\}\; =\; A^r.$
• $A^\{\backslash circ\}$ is closed in $Y$ under the weak-*-topology on $Y$.
• A subset $S$ of $X$ is weakly bounded (i.e. $\backslash sigma(X,\; Y)$-bounded) if and only if $S^\{\backslash circ\}$ is absorbing in $Y$.{{sfn|Narici|Beckenstein|2011|pp=225-273}}
• For a dual pair $\backslash langle\; X,\; X^\{\backslash prime\}\; \backslash rangle,$ where $X$ is a TVS and $X^\{\backslash prime\}$ is its continuous dual space, if $B\; \backslash subseteq\; X$ is bounded then $B^\{\backslash circ\}$ is absorbing in $X^\{\backslash prime\}.${{sfn|Trèves|2006|pp=195-201}} If $X$ is locally convex and $B^\{\backslash circ\}$ is absorbing in $X^\{\backslash prime\}$ then $B$ is bounded in $X.$ Moreover, a subset $S$ of $X$ is weakly bounded if and only if $S^\{\backslash circ\}$ is absorbing in $X^\{\backslash prime\}.$
• The bipolar $A^\{\backslash circ\backslash circ\}$ of a set $A$ is the $\backslash sigma(X,\; Y)$-closed convex hull of $A\; \backslash cup\; \backslash \{0\backslash \},$ that is the smallest $\backslash sigma(X,\; Y)$-closed and convex set containing both $A$ and $0.$
• Similarly, the bidual cone of a cone $A$ is the $\backslash sigma(X,\; Y)$-closed conic hull of $A.${{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}}
• If $\backslash mathcal\{B\}$ is a base at the origin for a TVS $X$ then $X^\{\backslash prime\}\; =\; \backslash bigcup\_\{B\; \backslash in\; \backslash mathbb\{B\}\}\; \backslash left(B^\{\backslash circ\}\backslash right).${{sfn|Narici|Beckenstein|2011|p=472}}
• If $X$ is a locally convex TVS then the polars (taken with respect to $\backslash left\backslash langle\; X,\; X^\{\backslash prime\}\; \backslash right\backslash rangle$) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of $X^\{\backslash prime\}$ (i.e. given any bounded subset $H$ of $X^\{\backslash prime\}\_\{\backslash sigma\},$ there exists a neighborhood $S$ of the origin in $X$ such that $H\; \backslash subseteq\; S^\{\backslash circ\}$).{{sfn|Schaefer|Wolff|1999|pp=123–128}}
• Conversely, if $X$ is a locally convex TVS then the polars (taken with respect to $\backslash langle\; X,\; X^\{\backslash \#\}\; \backslash rangle$) of any fundamental family of equicontinuous subsets of $X^\{\backslash prime\}$ form a neighborhood base of the origin in $X.${{sfn|Schaefer|Wolff|1999|pp=123–128}}
• Let $X$ be a TVS with a topology $\backslash tau.$ Then $\backslash tau$ is a locally convex TVS topology if and only if $\backslash tau$ is the topology of uniform convergence on the equicontinuous subsets of $X^\{\backslash prime\}.${{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 $X$'s original topology.

Set relations

• $X^\{\backslash circ\}\; =\; X^$

  r

  = X^r = \{0\}{{sfn|Schaefer|Wolff|1999|pp=123–128}} and $\backslash varnothing^\{\backslash circ\}\; =\; \backslash varnothing^$

  r

  = \varnothing^r = Y.
• For all scalars $s\; \backslash neq\; 0,$ $(s\; A)^\{\backslash circ\}\; =\; \backslash tfrac\{1\}\{s\}\; \backslash left(A^\{\backslash circ\}\backslash right)$ and for all real $t\; \backslash neq\; 0,$ $(t\; A)^$

  r

  = \tfrac{1}{t} \left(A^

  r

  \right) and $(t\; A)^r\; =\; \backslash tfrac\{1\}\{t\}\; \backslash left(A^r\backslash right).$
• $A^\{\backslash circ\backslash circ\backslash circ\}\; =\; A^\{\backslash circ\}.$ However, for the real polar we have $A^\{r\; r\; r\}\; \backslash subseteq\; A^r.${{sfn|Schaefer|Wolff|1999|pp=123–128}}
• For any finite collection of sets $A\_1,\; \backslash ldots,\; A\_n,$ $$\backslash left(A\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; A\_n\backslash right)^\{\backslash circ\}\; =\; \backslash left(A\_1^\{\backslash circ\}\backslash right)\; \backslash cup\; \backslash cdots\; \backslash cup\; \backslash left(A\_n^\{\backslash circ\}\backslash right).$$
• If $A\; \backslash subseteq\; B$ then $B^\{\backslash circ\}\; \backslash subseteq\; A^\{\backslash circ\},$ $B^r\; \backslash subseteq\; A^r,$ and $B^$

  r

  \subseteq A^

  r

  .
• An immediate corollary is that $\backslash bigcup\_\{i\; \backslash in\; I\}\; \backslash left(A\_i^\{\backslash circ\}\backslash right)\; \backslash subseteq\; \backslash left(\backslash bigcap\_\{i\; \backslash in\; I\}\; A\_i\backslash right)^\{\backslash circ\}$; equality necessarily holds when $I$ is finite and may fail to hold if $I$ is infinite.
• $\backslash bigcap\_\{i\; \backslash in\; I\}\; \backslash left(A\_i^\{\backslash circ\}\backslash right)\; =\; \backslash left(\backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i\backslash right)^\{\backslash circ\}$ and $\backslash bigcap\_\{i\; \backslash in\; I\}\; \backslash left(A\_i^r\backslash right)\; =\; \backslash left(\backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i\backslash right)^r.$
• If $C$ is a cone in $X$ then $C^\{\backslash circ\}\; =\; \backslash left\backslash \{\; y\; \backslash in\; Y\; :\; \backslash langle\; c,\; y\; \backslash rangle\; =\; 0\; \backslash text\{\; for\; all\; \}\; c\; \backslash in\; C\; \backslash right\backslash \}.${{sfn|Trèves|2006|pp=195-201}}
• If $\backslash left(S\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a family of $\backslash sigma(X,\; Y)$-closed subsets of $X$ containing $0\; \backslash in\; X,$ then the real polar of $\backslash cap\_\{i\; \backslash in\; I\}\; S\_i$ is the closed convex hull of $\backslash cup\_\{i\; \backslash in\; I\}\; \backslash left(S\_i^r\backslash right).${{sfn|Schaefer|Wolff|1999|pp=123–128}}
• If $0\; \backslash in\; A\; \backslash cap\; B$ then $A^\{\backslash circ\}\; \backslash cap\; B^\{\backslash circ\}\; \backslash subseteq\; 2\; \backslash left[(A\; +\; B)^\{\backslash circ\}\backslash right]\; \backslash subseteq\; 2\backslash left(A^\{\backslash circ\}\; \backslash cap\; B^\{\backslash circ\}\backslash right).${{sfn|Jarchow|1981|pp=148-150}}
• For a closed convex cone $C$ in a real vector space $X,$ the polar cone is the polar of $C$; that is, $$C^\{\backslash circ\}\; =\; \backslash \{y\; \backslash in\; Y\; :\; \backslash sup\_\{\}\; \backslash langle\; C,\; y\; \backslash rangle\; \backslash leq\; 0\backslash \},$$ where $\backslash sup\_\{\}\; \backslash langle\; C,\; y\; \backslash rangle\; :=\; \backslash sup\_\{c\; \backslash in\; C\}\; \backslash langle\; c,\; y\; \backslash rangle.${{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}}

• {{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}}

{{reflist|group=note}}

{{reflist}}

• {{Jarchow Locally Convex Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{Duality and spaces of linear maps}}

{{Topological vector spaces}}

{{Functional analysis}}

Category:Functional analysis

Category:Linear functionals

Category:Topological vector spaces