Reference request: complex half spaces

Question

For real spaces, the notion of a half space is well-known. It's given by the set $\{ x \in \mathbb{R}^n \, \mid \, y^\top x \leq b \}$ for some $y \in \mathbb{R}^n$ and $b \in \mathbb{R}$. For the space $\mathbb{C}^n$, a half-space should be the following $$\{ x \in \mathbb{C}^n \, | \, \text{Re}(y^\text{H} x) \leq b \},$$ where $y \in \mathbb{C}^n$, and $(\cdot)^\text{H}$ denotes the conjugate transpose. However, I cannot find any mention of this in the literature. Am I perhaps using the wrong terminology? For example, this post: Regarding closed half spaces in $\mathbb{C}^n$, mentions closed complex half-spaces, but I am unable to find the mentioned paper.

Answer 1

Your definition of a complex half-space coincides with what is used in the literature to formulate separation theorems (Hahn-Banach theorems): That is, separation of sets by half-spaces. See, e.g., Rudin, Functional analysis, Chapter 3. There exactly these sets are used, without calling them explicitly complex half spaces.