Dual cone and polar cone: Difference between revisions

Revision as of 22:21, 22 May 2020

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If ''X'' is a [[topological vector space]] over the real or complex numbers, then the '''dual cone''' of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'':

:<math>C^{\prime} := \left\{ f \in X^{\prime} : \operatorname{Re} \left( f (x) \right) \geq 0 \text{ for all } x \in C \right\}</math>.,{{sfn | Schaefer | 1999 | pp=215–222}}

which is the [[polar set|polar]] of the set -''C''.{{sfn | Schaefer | 1999 | pp=215–222}}

No matter what ''C'' is, <math>C^{\prime}</math> will be a convex cone.

If ''C'' ⊆ {0} then <math>C^{\prime} = X^{\prime}</math>.