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Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as '''R'''<sup>''n''</sup> equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.
:<math>C^*_\text{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}.</math>
Using this latter definition for ''C*'', we have that when ''C'' is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization | first1=Stephen P. |last1=Boyd |first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3 | url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref>
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