Page "Complex projective space" Paragraph 1

In mathematics, complex projective space is the projective space with respect to the field of complex numbers.

By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space ( see below for an intuitive account ).

Formally, a complex projective space is the space of complex lines through the origin of an ( n + 1 )- dimensional complex vector space.

The space is denoted variously as P ( C < sup > n + 1 </ sup >), P < sub > n </ sub >( C ) or CP < sup > n </ sup >.

When, the complex projective space CP < sup > 1 </ sup > is the Riemann sphere, and when, CP < sup > 2 </ sup > is the complex projective plane ( see there for a more elementary discussion ).