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.