Definition 4

Given a left (resp. right) vector space $V$ over a field $K$, the left (resp. right) projective space derived from $V$, denoted by $\mathbf{P}(V)$, is the quotient of the complement $V - \{0\}$ of $\{0\}$ in $V$ by the equivalence relation $\Delta(V)$ "there exists $\lambda \neq 0$ in $K$ such that $y = \lambda x$ (resp. $y = x \lambda$ ) between $x$ and $y$ in $V - \{0\}$.

When $V = K_s^{n+1}$, we also write $\mathbf{P}_n(K)$ instead of $\mathbf{P}(K_s^{n+1})$ and $\Delta_n(K)$ instead of $\Delta(V)$.

Definition 4 can also be expressed by saying that $\mathbf{P}(V)$ is the set of lines (passing through 0) in $V$ with the origin removed; $\mathbf{P}(V)$ is therefore canonically identified with the set of lines (passing through 0) in $V$. The elements of a projective space are called the points of that space.

When $V$ is of dimension $n$, the integer $n - 1$ is called the dimension of the projective space $\mathbf{P}(V)$ if $n$ is finite, and the cardinal $n$ otherwise; this cardinal is denoted by $\dim_K \mathbf{P}(V)$ or $\dim \mathbf{P}(V)$. Thus a projective space of dimension $-1$ is empty and a projective space of dimension 0 is a single point. A projective space of dimension 1 (resp. 2) is called a projective line (resp. projective plane).

Henceforth we shall only consider left projective spaces.