Definition:Projective Space

Definition
Let $K$ be a field.

Let $n \ge 0$ be a natural number.

Let $\sim$ be the equivalence relation defined on the set $K^{n+1} \setminus \left\{{0}\right\}$ by:


 * $x, y \in K^{n+1} \setminus \left\{{0}\right\}: x \sim y \iff \exists \lambda \in K: x = \lambda y$

The projective space of dimension $n$ over $K$ is the quotient set $\left({K^{n+1} \setminus \left\{{0}\right\}}\right) / \sim$ and is denoted:
 * $K \mathbb P^n$

or:
 * $\mathbb P^n \left({ K }\right)$

or:
 * $\mathbb P \left({ K^{n+1} }\right)$