Definition:Projective Space/Over a Field

Definition

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.

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