In the book Notions of Convexity by Lars Hörmander page 290, section 4.6 Linear convexity is defined as follows. An open set $X\in \mathbb{C}^n$ is called linearly convex if for every $z\in \mathbb{C}^n\setminus X $ there exists an affine complex hyperplane $\Pi$ such that $z\in \Pi\subset \mathbb{C}^n\setminus X$.
Proposition 4.6.2 says. If $X$ is an open set in $\mathbb{C}^n$, then the union $F$ Of all affine complex hyperplanes $\Pi\subset\mathbb{C}^n\setminus X$ is a closed set and $\hat{X}= \mathbb{C}^n\setminus F$ is linearly convex.
If $V$ is a complex vector space we shall denote by $P(V)$ the projective space consisting of all complex lines through the origin in $V$. And $V^*$ is the dual space of $V$.
I wanted to know that why did we need to go to the projective geometry set up to define $X^{**}$? Can we not define it in vector space set up itself? Do we have a similar definition of $X^{**}$ where we need not consider the projective space (something’s similar to the bipolar of a set in Banach space).
I have no background in projective geometry, but have started learning and have understood the real projective plane. Since I am not very sure of my concepts in projective geometry, I wanted to know if we have similar definition of $X^{**}$ in vector space set up.
Reference request: The most standard known books to study the different notions of convexity in several complex variables are Complex Convexity and Analytic Functionals by Mats Andersson and Notions of Convexity by Lars Hörmander. Are there any other references/texts with all equivalent definitions to study the same?