# convex cone in complex Banach space

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.

• Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces. Nov 22 '18 at 9:34

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 $$sx+ty \in C$$.