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2011-04-02T14:04:07Z

Created page with "==Definition== ===For a vector space over a field, as a set=== Suppose <math>k</math> is a field and <math>V</math> is a nonzero vector space over <math>k</math>. The '''pr..."

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==Definition==

===For a vector space over a field, as a set===

Suppose <math>k</math> is a [[field]] and <math>V</math> is a nonzero vector space over <math>k</math>. The '''projective space''' <math>\mathbb{P}(V)</math> is defined in the following equivalent ways:

* As the quotient of the set of nonzero elements of <math>V</math> under the equivalence relation of being in the same orbit under the action of the multiplicative group <math>k^*</math>.
* As the set of one-dimensional subspaces of <math>V</math>, i.e., the set of lines through the origin in <math>V</math>.
* As the set of codimension one subspaces of <math>V</math>, 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 <math>D</math> is a [[division ring]] and <math>V</math> is a nonzero left vector space over <math>D</math>. The '''projective space''' <math>\mathbb{P}(V)</math> is defined in the following equivalent ways:

* As the quotient of the set of nonzero elements of <math>V</mah> under the equivalence relation of being in the same orbit under the action of <math>D^*</math> under ''left'' multiplication.
* As the set of one-dimensional ''left'' vector subspaces of <math>V</math>, i.e., the set of lines through the origin in <math>V</math>.
* As the set of codimension one ''left'' vector subspaces of <math>V</math>, i.e., the set of subspaces for which the quotient space is one-dimensional.

===For a field or division ring and a parameter===

Suppose <math>k</math> is a [[field]] or [[division ring]] and <math>n</math> is a nonnegative integer. The '''projective space''' <math>\mathbb{P}^n(k)</math> is defined as the projective space corresponding to the vector space <math>k^{n+1}</math>. 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 <math>k</math> is a [[topological field]] or [[topological division ring]] and <math>n</math> is a nonnegative integer. The '''projective space''' <math>\mathbb{P}^n(k)</math> now has the structure of a ''topological'' space as follows: We first equip <math>k^{n+1}</math> with the [[product topology]] arising from <math>k</matH>. We then equip <math>k^{n+1} \setminus \{ 0 \}</math> with the [[subspace topology]] arising from <math>k^{n+1}</math>. Finally, we equip the quotient under the action of <math>k^*</math> 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 <math>k^{n+1}</math> for some <math>n</math>.

===For an infinite-dimensional topological vector space===

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

Projective space - Revision history

Vipul: Created page with "==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 '''pr..."

Vipul: Created page with "==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 '''pr..."