Projective subspace

Summary

A projective subspace of a projective space $\mathbb{P}(V)$ is the set of points that come from a linear subspace of $V$. Points, lines and planes of a projective space are its projective subspaces of dimension $0$, $1$ and $2$. Equivalently, it is a set of points that contains the whole line through any two of its points. The projective subspaces form a lattice that is isomorphic to the lattice of linear subspaces of $V$.