Definition:Projective Space

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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 $V \setminus \left\{{0}\right\}$ by:

$x, y \in V \setminus \left\{{0}\right\}: x \sim y \iff \exists \lambda \in K: x = \lambda y$

The projective space associated to $V$ of dimension $n$ over $K$ is the quotient set $\left({V \setminus \left\{{0}\right\}}\right) / \sim$ and is denoted $\mathbb P \left({ V }\right)$.

If $V = K^{n+1}$ for $n \ge 0$ a natural number, projective space is sometimes denoted $\mathbb P\left( K^{n+1} \right) = \mathbb P^n\left( K \right)$.

This is because while $K^{n+1}$ is an $\left(n+1\right)$-dimensional vector space, the projective space $\mathbb P\left( K^{n+1} \right)$ has dimension $n$.

The notation $\mathbb P\left( K^{n+1} \right) = K\mathbb P^n$ is also in use.