Dual cone and polar cone: Difference between revisions

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Revision as of 16:22, 26 March 2012

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For a set <math>C</math> in <math>X</math>, the '''polar cone''' of <math>C</math> is the set

:<math>C^o = \left \{y\in X^*: \langle y , x \rangle \leq 10 \quad \forall x\in C \right \}.</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>

ForIt acan closedbe convexseen cone <math>C</math> in <math>X</math>,that the polar cone cone is equal to the negative of the dual cone, i.e. <math>C^o=-C^*</math>.

For a closed convex cone <math>C</math> in <math>X</math>, the polar cone is equivalent to the [[polar set]] for <math>C</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>

== See also ==