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A convex cone is defined as (by Wikipedia): A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

In my research work, I need a convex cone in a complex Banach space, but the set of complex numbers is not an ordered field. Then how to define a convex cone in a complex Banach space? I tried to define such a partial order on $\mathbb C$ so that it can be a total order on $\mathbb C$ but I could not succeed.

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    $\begingroup$ Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces. $\endgroup$ Nov 22 '18 at 9:34
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If $X$ is a complex Banach space and $C \subseteq X$, then $C$ is called a convex cone if $x,y \in C $ and $s,t \in [0, \infty)$ imply that $sx+ty \in C$.

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