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?
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&nbsp;p. (2004). ZBL1050.51001.
Projective geometry is the study of projective spaces.