Projective space

Definition

For a vector space over a field, as a set

Suppose $k$ is a field and $V$ is a nonzero vector space over $k$. The projective space $\mathbb {P} (V)$ is defined in the following equivalent ways:

• As the quotient of the set of nonzero elements of $V$ under the equivalence relation of being in the same orbit under the action of the multiplicative group $k^{*}$.
• As the set of one-dimensional subspaces of $V$, i.e., the set of lines through the origin in $V$.
• As the set of codimension one subspaces of $V$, i.e., the set of subspaces for which the quotient space is one-dimensional.

For a left vector space over a division ring, as a set

Suppose $D$ is a division ring and $V$ is a nonzero left vector space over $D$. The projective space $\mathbb {P} (V)$ is defined in the following equivalent ways:

• As the quotient of the set of nonzero elements of $V</mah>undertheequivalencerelationofbeinginthesameorbitundertheactionof<math>D^{*}$ under left multiplication.
• As the set of one-dimensional left vector subspaces of $V$, i.e., the set of lines through the origin in $V$.
• As the set of codimension one left vector subspaces of $V$, i.e., the set of subspaces for which the quotient space is one-dimensional.

For a field or division ring and a parameter

Suppose $k$ is a field or division ring and $n$ is a nonnegative integer. The projective space $\mathbb {P} ^{n}(k)$ is defined as the projective space corresponding to the vector space $k^{n+1}$. Note that the corresponding vector space has dimension one more than the parameter (and also the algebraic dimension) for the projective space, and the reason for this is that when we quotient to the set of orbits under the action of the multiplicative group, we are destroying one dimension.

For a topological field or division ring and a parameter

Suppose $k$ is a topological field or topological division ring and $n$ is a nonnegative integer. The projective space $\mathbb {P} ^{n}(k)$ now has the structure of a topological space as follows: We first equip $k^{n+1}$ with the product topology arising from $k$. We then equip $k^{n+1}\setminus \{0\}$ with the subspace topology arising from $k^{n+1}$. Finally, we equip the quotient under the action of $k^{*}$ with the quotient topology.

Note that this topologization works, more generally, for any projective space corresponding to a finite-dimensional vector space, because we can identify the vector space with $k^{n+1}$ for some $n$.

For an infinite-dimensional topological vector space

Suppose $k$ is a topological field or topological division ring and $V$ is a (possibly infinite-dimensional) topological vector space over $k$. The projective space $\mathbb {P} (V)$ has the structure of a topological space as follows: first, $V\setminus \{0\}$ gets the subspace topology from $V$, then the quotient under the action of $k^{*}$ gets the quotient topology.