Editing Projective space (section)

== Definition ==
Given a [[vector space]] {{mvar|V}} over a [[field (mathematics)|field]] {{mvar|K}}, the ''projective space'' {{math|'''P'''(''V'')}} is the set of [[equivalence class]]es of {{math|''V'' \ {{mset|0}}}} under the equivalence relation {{math|~}} defined by {{math|''x'' ~ ''y''}} if there is a nonzero element {{mvar|λ}} of {{mvar|K}} such that {{math|1=''x'' = ''λy''}}. If {{mvar|V}} is a [[topological vector space]], the quotient space {{math|'''P'''(''V'')}} is a [[topological space]], endowed with the [[quotient topology]] of the [[subspace topology]] of {{math|''V'' \ {{mset|0}}}}. This is the case when {{mvar|K}} is the field {{math|'''R'''}} of the [[real number]]s or the field {{math|'''C'''}} of the [[complex number]]s. If {{mvar|V}} is finite dimensional, the ''dimension'' of {{math|'''P'''(''V'')}} is the dimension of {{mvar|V}} minus one.

In the common case where {{math|1=''V'' = ''K''{{sup|''n''+1}}}}, the projective space {{math|'''P'''(''V'')}} is denoted {{math|'''P'''{{sub|''n''}}(''K'')}} (as well as {{math|''K'''''P'''{{sup|''n''}}}} or {{math|'''P'''{{sup|''n''}}(''K'')}}, although this notation may be confused with exponentiation). The space {{math|'''P'''{{sub|''n''}}(''K'')}} is often called ''the'' projective space of dimension {{mvar|n}} over {{mvar|K}}, or ''the projective {{mvar|n}}-space'', since all projective spaces of dimension {{mvar|n}} are [[isomorphism|isomorphic]] to it (because every {{mvar|K}} vector space of dimension {{math|''n'' + 1}} is isomorphic to {{math|''K''{{sup|''n''+1}}}}).

The elements of a projective space {{math|'''P'''(''V'')}} are commonly called ''[[Point (geometry)|points]]''. If a [[basis (vector space)|basis]] of {{mvar|V}} has been chosen, and, in particular if {{math|1=''V'' = ''K''{{sup|''n''+1}}}}, the [[projective coordinates]] of a point ''P'' are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted {{math|[''x''<sub>0</sub> : ... : ''x''<sub>''n''</sub>]}}, the colons and the brackets being used for distinguishing from usual coordinates, and emphasizing that this is an equivalence class, which is defined [[up to]] the multiplication by a non zero constant. That is, if {{math|[''x''<sub>0</sub> : ... : ''x''<sub>''n''</sub>]}} are projective coordinates of a point, then {{math|[''λx''<sub>0</sub> : ... : ''λx''<sub>''n''</sub>]}} are also projective coordinates of the same point, for any nonzero {{mvar|λ}} in {{mvar|K}}. Also, the above definition implies that {{math|[''x''<sub>0</sub> : ... : ''x''<sub>''n''</sub>]}} are projective coordinates of a point if and only if at least one of the coordinates is nonzero.

If {{mvar|K}} is the field of real or complex numbers, a projective space is called a [[real projective space]] or a [[complex projective space]], respectively. If {{math|''n''}} is one or two, a projective space of dimension {{math|''n''}} is called a [[projective line]] or a [[projective plane]], respectively. The complex projective line is also called the [[Riemann sphere]].

All these definitions extend naturally to the case where {{mvar|K}} is a [[division ring]]; see, for example, ''[[Quaternionic projective space]]''. The notation {{math|PG(''n'', ''K'')}} is sometimes used for {{math|'''P'''{{sub|''n''}}(''K'')}}.<ref>Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', p.&nbsp;506, [[Marcel Dekker]] {{isbn|0-8247-0609-9}}</ref> If {{mvar|K}} is a [[finite field]] with {{mvar|q}} elements, {{math|'''P'''{{sub|''n''}}(''K'')}} is often denoted {{math|PG(''n'', ''q'')}} (see ''[[PG(3,2)]]'').{{efn|The absence of space after the comma is common for this notation.}}