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Dual cone and polar cone are defined only on $\mathbb R^n$. Has anyone seen the extension to $\mathbb C^n$? Any references for these?

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Are you just wondering or you're asking if someone else already has defined such a thing for a purpose? Because there is probably a way to define such a generalization but I've never heard of a purpose for it, though. – Patrick Da Silva Dec 18 '11 at 23:53
I am wondering if someone else already has defined such a thing for a purpose. – Sunni Dec 19 '11 at 0:18
Very good question then, I am curious too. Would love to see an answer ; +1! – Patrick Da Silva Dec 19 '11 at 0:25
The definition on Wikipedia automatically generalizes to any inner product space (and $\mathbb{C}$ has of course an inner product) – Fredrik Meyer Dec 19 '11 at 6:04
@Fredrik, you mean to take real part of inner product of complex vectors? – Sunni Dec 19 '11 at 15:28

As I see it, the definition in Wikipedia is for any Linear Space. The book by Aliprantis and Border defines polars for any Dual Set, which is even more general. These general definitions are very useful in Convex Optimization, but I've never seen it used on $\mathbb{C}^n$.

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