I am only learning convex analysis properly now for the first time, and most of the references I am using only deal with topological vector spaces over $\mathbb{R}$. Is there any serious stumbling block to simply generalizing everything to complex vector spaces?
To give a concrete example in a finite-dimensional case, normally if we have a closed convex cone $K$ that is not all of $\mathbb{R}^n$, then the hyperplane separation theorem can be stated as saying that for every $c \notin K$, there exists $v \in \mathbb{R}^n$ such that $v^T c < 0$ and $v^T k \geq 0$ for all $k \in K$.
Are there any obstructions to generalizing this as follows: for a closed convex cone $K$ that is not all of $\mathbb{C}^n$, then for every $c \notin K$ there exists $v \in \mathbb{C}^n$ such that $\mathrm{Re}(v^*c) < 0$ and $\mathrm{Re}(v^* k) \geq 0$ for all $k \in K$. The result appears to be true by applying the complex Hahn-Banach theorem to the finite-dimensional case, unless I'm mistaken.
