A projective space is a space that is invariant under the group 
of all general linear homogeneous transformation in the space concerned, but not under all the transformations of any group containing 
as a subgroup.
A projective space is the space of one-dimensional vector subspaces of a given vector space. For real vector spaces, the notation 
or 
denotes the real projective space of dimension 
(i.e., the space of one-dimensional vector subspaces of 
) and 
denotes the complex projective space of complex dimension 
(i.e., the space of one-dimensional complex vector subspaces of 
). 
can also be viewed as the set consisting of 
together with its points at infinity.
