projective geometry and projective space Let $V$ is a vectorspace  over field $F_q$, we denote the set of all subspaces of $V$ by $\mathcal{P}(V)$. I saw some referencess they mentioned $\mathcal{P}(V)$ as a projective space and some referencess they mentioned as projective geometry. what is the difference between a projective geometry and a projective space?
 A: Geometry is defined as structure $G=(\Omega,I)$ consisting of set $\Omega$ and a relation I. That definition is pretty abstract and can be used on whole variety of objects and relations. 
Once you get points and lines as elements of $\Omega$ and incidence relation for I, you get projective geometry.
Now we introduce axioms of projective geometry (which I am not going to write here) and we can define projective space as geometry $G=(\Omega,I)$ which satisfies given axioms.
Beutelspacher, Albrecht; Rosenbaum, Ute, Projective geometry. From foundations to applications, Vieweg Studium 41. Aufbaukurs Mathematik. Braunschweig: Vieweg (ISBN 3-528-17241-X/pbk). x, 265 p. (2004). ZBL1050.51001. 
A: It likes the difference between the number theory and the number field. Projective geometry is not a mathematic concept while projective space is a strictly defined object i.e. $(\mathbb P^n, V,\pi )$ where $\mathbb P^n$ is a set, $V$ is a $n+1$ dimensions vector space, and  $\pi $ is a projection from  $V-\{0\}\to \mathbb P^n$ satisfying several laws (homogeneity etc). You can find its detailed definition in any modern textbook(always the first chapter of algebraic geometry tutorial). 
A: Projective geometry is the study of projective spaces.
