Definition:Projective Space
Definition
Real Projective Space
Let $\Bbb S^n \subseteq \R^{n+1}$ be an $n$-sphere.
Let $\sim$ be the equivalence relation defined on $\Bbb S^n$ by:
- $x, y \in \Bbb S^n: x \sim y \iff x = -y$
The real projective space of dimension $n$ is the quotient space $\Bbb S^n / \sim$ and is denoted $\Bbb{RP}^n$.
Over a Field
Let $V$ be a vector space over a field $K$ of dimension $n + 1 \ge 1$.
Let $\sim$ be the equivalence relation defined on the set $V \setminus \set 0$ by:
- $x, y \in V \setminus \set 0: x \sim y \iff \exists \lambda \in K: x = \lambda y$
The projective space associated to $V$ of dimension $n$ over $K$ is the quotient set $\paren {V \setminus \set 0} / \sim$ and is denoted $\map {\mathbb P} V$.
If $V = K^{n + 1}$ for $n \ge 0$ a natural number, projective space is sometimes denoted $\map {\mathbb P} {K^{n + 1} } = \map {\mathbb P^n} K$.
This is because while $K^{n + 1}$ is an $\paren {n + 1}$-dimensional vector space, the projective space $\map {\mathbb P} {K^{n + 1} }$ has dimension $n$.
The notation $\map {\mathbb P} {K^{n + 1} } = K \mathbb P^n$ is also in use.