Projective subspace

Remarks

Equivalently, a set $S$ of points is a projective subspace when, for all $[v], [w] \in S$ and all scalars $a, b$ with $a v + b w \ne 0$, the point $[a v + b w]$ lies in $S$. In words, $S$ contains the whole line through any two of its distinct points.

Example: in $\mathbb{P}^2(\R)$, the set $\set{[x_0 : x_1 : x_2] : x_0 = 0}$ is a projective subspace of dimension $1$, a line. Check: it is $\mathbb{P}(W)$ for the plane $W = \set{x_0 = 0}$ of $\R^3$, which has dimension $2$.

Non-example: the set $\set{[1:0:0], [0:1:0]}$ is not a projective subspace. The line through its two points also contains $[1:1:0]$, which is not in the set.

The empty set is $\mathbb{P}(0)$, of dimension $-1$, and $\mathbb{P}(V)$ itself is a projective subspace. For finite-dimensional $V$, a projective subspace of dimension $\dim V - 2$ is a hyperplane.