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''C*'' is always a [[convex cone]], even if ''C'' is neither [[convex set|convex]] nor a [[linear cone|cone]].
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>
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