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| :''See also [[polar set (potential theory)]].''
| | Hello!<br><br>Allow me to start by saying my name - Lindsey Laforge. [http://search.about.com/?q=Connecticut Connecticut] is where my house is but I will have to handle in per year or two. Hiring has been her regular job for years but soon she'll be on her. Her husband doesn't like it the way she does but what she really loves doing is astrology but she's been taking on new things lately. You can always find his [http://Pinterest.com/search/pins/?q=website website] here: http://www.pinterest.com/seodress/rocklin-real-estate-qualified-realtor-in-the-rockl/ |
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| In [[functional analysis]] and related areas of [[mathematics]] the '''polar set''' of a given subset of a [[vector space]] is a certain set in the [[dual space]].
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| Given a [[dual pair]] <math>(X,Y)</math> the '''polar set''' or '''polar''' of a subset <math>A</math> of <math>X</math> is a set <math>A^\circ</math> in <math>Y</math> defined as
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| :<math>A^\circ := \{y \in Y : \sup_{x \in A} |\langle x,y \rangle | \le 1\}</math>
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| The '''bipolar''' of a subset <math>A</math> of <math>X</math> is the polar of <math>A^\circ</math>. It is denoted <math>A^{\circ\circ}</math> and is a set in <math>X</math>.
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| == Properties ==
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| * <math>A^\circ</math> is [[absolutely convex]]
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| * If <math>A \subseteq B</math> then <math>B^\circ \subseteq A^\circ</math>
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| * For all <math>\gamma \neq 0</math> : <math>(\gamma A)^\circ = \frac{1}{\mid\gamma\mid}A^\circ</math>
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| * <math>(\bigcup_{i \in I} A_i)^\circ = \bigcap_{i \in I}A_i^\circ</math>
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| * For a dual pair <math>(X,Y)</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>
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| * The bipolar <math>A^{\circ\circ}</math> of a set <math>A</math> is the [[absolutely convex envelope]] of <math>A</math>, that is the smallest absolutely convex set containing <math>A</math>. If <math>A</math> is already absolutely convex then <math>A^{\circ\circ}=A</math>.
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| * For a closed [[convex cone]] <math>C</math> in <math>X</math>, the [[polar cone]] is equivalent to the one-sided polar set for <math>C</math>, given by
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| : <math>C^\circ = \{y \in Y : \sup\{\langle x,y \rangle : x \in C \} \le 1\}</math>.<ref>{{cite book|last=Aliprantis|first=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>
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| == Geometry ==
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| In [[geometry]], the polar set may also refer to a duality between points and planes. In particular, the polar set of a point <math>x_0</math>, given by the set of points <math>x</math> satisfying <math>\langle x, x_0 \rangle=0</math> is its ''polar hyperplane,'' and the dual relationship for a hyperplane yields its ''pole.''
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| == See also ==
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| * [[Polar cone]]
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| * [[Bipolar theorem]]
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| == References ==
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| {{Refimprove|date=March 2012}}
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| {{Reflist}}
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| [[Category:Functional analysis]]
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| {{Mathanalysis-stub}}
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Hello!
Allow me to start by saying my name - Lindsey Laforge. Connecticut is where my house is but I will have to handle in per year or two. Hiring has been her regular job for years but soon she'll be on her. Her husband doesn't like it the way she does but what she really loves doing is astrology but she's been taking on new things lately. You can always find his website here: http://www.pinterest.com/seodress/rocklin-real-estate-qualified-realtor-in-the-rockl/