Editing Projective space

{{short description|Completion of the usual space with "points at infinity"}}
[[File:Railroad-Tracks-Perspective.jpg|thumb|right|In [[perspective (graphical)|graphical perspective]], parallel (horizontal) lines in the plane intersect at a [[vanishing point]] (on the [[horizon]]).]]
In [[mathematics]], the concept of a '''projective space''' originated from the visual effect of [[perspective (graphical)|perspective]], where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a [[Euclidean space]], or, more generally, an [[affine space]] with [[points at infinity]], in such a way that there is one point at infinity of each [[affine space#direction|direction]] of [[parallel lines]].

This definition of a projective space has the disadvantage of not being [[isotropic]], having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In [[synthetic geometry]], ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the [[axioms of projective geometry]]. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.

Using [[linear algebra]], a projective space of dimension {{mvar|n}} is defined as the set of the [[vector line]]s (that is, vector subspaces of dimension one) in a [[vector space]] {{mvar|V}} of dimension {{math|''n'' + 1}}. Equivalently, it is the [[quotient set]] of {{math|''V'' \ {{mset|0}}}} by the [[equivalence relation]] "being on the same vector line". As a vector line intersects the [[unit sphere]] of {{mvar|V}} in two [[antipodal points]], projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a [[projective line]], and a projective space of dimension 2 is a [[projective plane]].

Projective spaces are widely used in [[geometry]], allowing for simpler statements and simpler proofs. For example, in [[affine geometry]], two distinct lines in a plane intersect in at most one point, while, in [[projective geometry]], they intersect in exactly one point. Also, there is only one class of [[conic section]]s, which can be distinguished only by their intersections with the line at infinity: two intersection points for [[hyperbola]]s; one for the [[parabola]], which is tangent to the line at infinity; and no real intersection point of [[ellipse]]s.

In [[topology]], and more specifically in [[manifold theory]], projective spaces play a fundamental role, being typical examples of [[non-orientable manifold]]s.

== Motivation ==
[[File:Affine space R3.png|thumb|Projective plane and [[central projection]]]]
As outlined above, projective spaces were introduced for formalizing statements like "two [[coplanar lines]] intersect in exactly one point, and this point is at infinity if the lines are [[parallel lines|parallel]]". Such statements are suggested by the study of [[perspective (graphical)|perspective]], which may be considered as a [[central projection]] of the [[three dimensional space]] onto a [[plane (geometry)|plane]] (see ''[[Pinhole camera model]]''). More precisely, the entrance pupil of a camera or of the eye of an observer is the ''center of projection'', and the image is formed on the ''projection plane''.

Mathematically, the center of projection is a point {{mvar|O}} of the space (the intersection of the axes in the figure); the projection plane ({{math|''P''{{sub|2}}}}, in blue on the figure) is a plane not passing through {{mvar|O}}, which is often chosen to be the plane of equation {{math|1=''z'' = 1}}, when [[Cartesian coordinates]] are considered. Then, the central projection maps a point {{mvar|P}} to the intersection of the line {{mvar|OP}} with the projection plane. Such an intersection exists if and only if the point {{mvar|P}} does not belong to the plane ({{math|''P''{{sub|1}}}}, in green on the figure) that passes through {{mvar|O}} and is parallel to {{math|''P''{{sub|2}}}}.

It follows that the lines passing through {{mvar|O}} split in two disjoint subsets: the lines that are not contained in {{math|''P''{{sub|1}}}}, which are in one to one correspondence with the points of {{math|''P''{{sub|2}}}}, and those contained in {{math|''P''{{sub|1}}}}, which are in one to one correspondence with the directions of parallel lines in {{math|''P''{{sub|2}}}}. This suggests to define the ''points'' (called here ''projective points'' for clarity) of the projective plane as the lines passing through {{mvar|O}}. A ''projective line'' in this plane consists of all projective points (which are lines) contained in a plane passing through {{mvar|O}}. As the intersection of two planes passing through {{mvar|O}} is a line passing through {{mvar|O}}, the intersection of two distinct projective lines consists of a single projective point. The plane {{math|''P''{{sub|1}}}}
defines a projective line which is called the ''line at infinity'' of {{math|''P''{{sub|2}}}}. By identifying each point of {{math|''P''{{sub|2}}}} with the corresponding projective point, one can thus say that the projective plane is the [[disjoint union]] of {{math|''P''{{sub|2}}}} and the (projective) line at infinity.

As an [[affine space]] with a distinguished point {{mvar|O}} may be identified with its associated [[vector space]] (see ''{{slink|Affine space|Vector spaces as affine spaces}}''), the preceding construction is generally done by starting from a vector space and is called [[projectivization]]. Also, the construction can be done by starting with a vector space of any positive dimension.

So, a projective space of dimension {{mvar|n}} can be defined as the set of [[vector line]]s (vector subspaces of dimension one) in a vector space of dimension {{math|''n'' + 1}}. A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.

This set can be the set of [[equivalence class]]es under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed.

A third equivalent definition is to define a projective space of dimension {{mvar|n}} as the set of pairs of [[antipodal points]] in a sphere of dimension {{mvar|n}} (in a space of dimension {{math|''n'' + 1}}).

== 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.}}

== Related concepts ==

=== Subspace ===
Let {{math|'''P'''(''V'')}} be a projective space, where {{mvar|V}} is a vector space over a field {{mvar|K}}, and 
<math display="block">p:V\to \mathbf P(V)</math>
be the ''canonical map'' that maps a nonzero vector {{mvar|v}} to its equivalence class, which is the [[vector line]] containing {{mvar|v}} with the zero vector removed.

Every [[linear subspace]] {{mvar|W}} of {{mvar|V}} is a union of lines. It follows that {{math|''p''(''W'')}} is a projective space, which can be identified with {{math|'''P'''(''W'')}}.

A ''projective subspace'' is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines {{math|'''P'''(''V'')}}.

If {{math|''p''(''v'')}} and {{math|''p''(''w'')}} are two different points of {{math|'''P'''(''V'')}}, the vectors {{mvar|v}} and {{mvar|w}} are [[linearly independent]]. It follows that:
* There is exactly one projective line that passes through two different points of {{math|'''P'''(''V'')}}, and 
* A subset of {{math|'''P'''(''V'')}} is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.

In [[synthetic geometry]], where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.

=== Span ===
Every [[set intersection|intersection]] of projective subspaces is a projective subspace. It follows that for every subset {{mvar|S}} of a projective space, there is a smallest projective subspace containing {{mvar|S}}, the intersection of all projective subspaces containing {{mvar|S}}. This projective subspace is called the ''projective span'' of {{mvar|S}}, and {{mvar|S}} is a spanning set for it.

A set {{mvar|S}} of points is ''projectively independent'' if its span is not the span of any proper subset of {{mvar|S}}. If {{mvar|S}} is a spanning set of a projective space {{mvar|P}}, then there is a subset of {{mvar|S}} that spans {{mvar|P}} and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of {{mvar|P}} is {{mvar|n}}, such an independent spanning set has {{math|''n'' + 1}} elements.

Contrarily to the cases of [[vector space]]s and [[affine space]]s, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.

=== Frame ===
{{main|Projective frame}}

A ''projective frame'' or ''projective basis'' is an ordered set of points in a projective space that allows defining coordinates.{{sfn|Berger|2009|loc=chapter 4.4. Projective bases}} More precisely, in an {{mvar|n}}-dimensional projective space, a projective frame is a tuple of {{math|''n'' + 2}} points such that any {{math|''n'' + 1}} of them are independent; that is, they are not contained in a [[hyperplane]].

If {{mvar|V}} is an {{math|(''n'' + 1)}}-dimensional vector space, and {{mvar|p}} is the canonical projection from {{mvar|V}} to {{math|'''P'''(''V'')}}, then {{math|(''p''(''e''<sub>0</sub>), ..., ''p''(''e''<sub>''n''+1</sub>))}} is a projective frame if and only if {{math|(''e''<sub>0</sub>, ..., ''e''<sub>''n''</sub>)}} is a basis of {{mvar|V}} and the coefficients of {{math|''e''<sub>''n''+1</sub>}} on this basis are all nonzero. By rescaling the first {{mvar|n}} vectors, any frame can be rewritten as {{math|(''p''(''e''′<sub>0</sub>), ..., p(''e''′<sub>''n''+1</sub>))}} such that {{math|1=''e''′<sub>''n''+1</sub> = ''e''′<sub>0</sub> + ... + ''e''′<sub>''n''</sub>}}; this representation is unique up to the multiplication of all {{math|''e''′<sub>''i''</sub>}} with a common nonzero factor.

The ''projective coordinates'' or ''homogeneous coordinates'' of a point {{math|''p''(''v'')}} on a frame {{math|(''p''(''e''<sub>0</sub>), ..., ''p''(''e''<sub>''n''+1</sub>))}} with {{math|1=''e''<sub>''n''+1</sub> = ''e''<sub>0</sub> + ... + ''e''<sub>''n''</sub>}} are the coordinates of {{mvar|v}} on the basis {{math|(''e''<sub>0</sub>, ..., ''e''<sub>''n''</sub>)}}. They are only defined up to scaling with a common nonzero factor.

The ''canonical frame'' of the projective space {{math|'''P'''{{sub|''n''}}(''K'')}} consists of images by {{mvar|p}} of the elements of the canonical basis of {{math|''K''{{sup|''n''+1}}}} (that is, the [[tuples]] with only one nonzero entry, equal to 1), and the image by {{mvar|p}} of their sum.

=== Projective geometry ===
{{excerpt|Projective geometry|templates=-General geometry}}

=== Projective transformation ===
{{excerpt|Homography}}

== Topology ==
{{hatnote|In this section, all projective spaces are real projective spaces of finite dimension. However everything applies to complex projective spaces, with slight modifications.}}

A projective space is a [[topological space]], as endowed with the [[quotient topology]] of the topology of a finite dimensional real vector space.

Let {{mvar|S}} be the [[unit sphere]] in a normed vector space {{mvar|V}}, and consider the function
<math display="block">\pi: S \to \mathbf P(V)</math>
that maps a point of {{mvar|S}} to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of {{math|'''P'''(''V'')}} consist of two [[antipodal point]]s. As spheres are [[compact space]]s, it follows that: 
{{block indent | em = 1.5 | text =''A (finite dimensional) projective space is compact''.}}

For every point {{mvar|P}} of {{mvar|S}}, the restriction of {{pi}} to a neighborhood of {{mvar|P}} is a [[homeomorphism]] onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple [[atlas (topology)|atlas]] can be provided, as follows.

As soon as a basis has been chosen for {{mvar|V}}, any vector can be identified with its coordinates on the basis, and any point of {{math|'''P'''(''V'')}} may be identified with its [[homogeneous coordinates]]. For {{math|1=''i'' = 0, ..., ''n''}}, the set
<math display="block">U_i = \{[x_0:\cdots: x_n], x_i \neq 0\}</math>
is an open subset of {{math|'''P'''(''V'')}}, and 
<math display="block">\mathbf P(V) = \bigcup_{i=0}^n U_i</math>
since every point of {{math|'''P'''(''V'')}} has at least one nonzero coordinate.

To each {{math|''U''{{sub|''i''}}}} is associated a [[chart (topology)|chart]], which is the [[homeomorphism]]s
<math display="block">\begin{align}
\mathbb \varphi_i: R^n &\to U_i\\
(y_0,\dots,\widehat{y_i},\dots, y_n)&\mapsto [y_0:\cdots:y_{i-1}:1:y_{i+1}:\cdots:y_n],
\end{align}</math>
such that 
<math display="block">\varphi_i^{-1}\left([x_0:\cdots:x_n]\right) =\left (\frac{x_0}{x_i}, \dots, \widehat{\frac{x_i}{x_i}}, \dots, \frac{x_n}{x_i} \right ),</math>
where hats means that the corresponding term is missing.
[[File:P1 ako varieta.png|thumb|200px|right|Manifold structure of the real projective line]]

These charts form an [[atlas (topology)|atlas]], and, as the [[transition map]]s are [[analytic function]]s, it results that projective spaces are [[analytic manifold]]s.

For example, in the case of {{math|1=''n'' = 1}}, that is of a projective line, there are only two {{math|''U''{{sub|''i''}}}}, which can each be identified to a copy of the [[real line]]. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is 
<math display="block">x\mapsto \frac 1 x</math>
in both directions. The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be identified with the projective line with a single point removed.

=== CW complex structure ===
Real projective spaces have a simple [[CW complex]] structure, as {{math|'''P'''<sup>''n''</sup>('''R''')}} can be obtained from {{math|'''P'''<sup>''n''−1</sup>('''R''')}} by attaching an {{math|''n''}}-cell with the quotient projection {{math|'''S'''<sup>''n''−1</sup> → '''P'''<sup>''n''−1</sup>('''R''')}} as the attaching map.

== Algebraic geometry ==
Originally, [[algebraic geometry]] was the study of common zeros of sets of [[multivariate polynomial]]s. These common zeros, called [[algebraic varieties]] belong to an [[affine space]]. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the [[fundamental theorem of algebra]] asserts that a univariate [[square-free polynomial]] of degree {{mvar|n}} has exactly {{mvar|n}} complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider ''zeros at infinity''. For example, [[Bézout's theorem]] asserts that the intersection of two plane [[algebraic curve]]s of respective degrees {{mvar|d}} and {{mvar|e}} consists of exactly {{mvar|de}} points if one consider complex points in the projective plane, and if one counts the points with their multiplicity.{{efn|The correct definition of the multiplicity if not easy and dates only from the middle of 20th century}} Another example is the [[genus–degree formula]] that allows computing the genus of a plane [[algebraic curve]] from its [[singular point of a curve|singularities]] in the ''complex projective plane''.

So a [[projective variety]] is the set of points in a projective space, whose [[homogeneous coordinates]] are common zeros of a set of [[homogeneous polynomial]]s.{{efn|Homogeneous required in order that a zero remains a zero when the homogeneous coordinates are multiplied by a nonzero scalar.}}

Any affine variety can be ''completed'', in a unique way, into a projective variety by adding its [[points at infinity]], which consists of [[homogenization of a polynomial|homogenizing]] the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by [[saturation (commutative algebra)|saturating]] with respect to the homogenizing variable.

An important property of projective spaces and projective varieties is that the image of a projective variety under a [[morphism of algebraic varieties]] is closed for [[Zariski topology]] (that is, it is an [[algebraic set]]). This is a generalization to every ground field of the compactness of the real and complex projective space.

A projective space is itself a projective variety, being the set of zeros of the zero polynomial.

=== Scheme theory ===
Scheme theory, introduced by [[Alexander Grothendieck]] during the second half of 20th century, allows defining a generalization of algebraic varieties, called [[scheme (mathematics)|schemes]], by gluing together smaller pieces called [[affine schemes]], similarly as [[manifold]]s can be built by gluing together open sets of {{math|'''R'''<sup>''n''</sup>}}.  The [[Proj construction]] is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold.
{{See also|Algebraic geometry of projective spaces}}

== Synthetic geometry ==
In [[synthetic geometry]], a '''projective space''' {{math|''S''}} can be defined axiomatically as a set {{math|''P''}} (the set of points), together with a set {{math|''L''}} of subsets of {{math|''P''}} (the set of lines), satisfying these axioms:<ref>{{harvnb|Beutelspacher|Rosenbaum|1998|pp=6–7}}</ref>
* Each two distinct points {{math|''p''}} and {{math|''q''}} are in exactly one line.
* [[Oswald Veblen|Veblen]]'s axiom:{{efn|also referred to as the ''Veblen–Young axiom'' and mistakenly as the [[Pasch's axiom|axiom of Pasch]] {{harv|Beutelspacher|Rosenbaum|1998|pp=6–7}}. Pasch was concerned with real projective space and was attempting to introduce order, which is not a concern of the Veblen–Young axiom.}} If {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} are distinct points and the lines through {{math|''ab''}} and {{math|''cd''}} meet, then so do the lines through {{math|''ac''}} and {{math|''bd''}}.
* Any line has at least 3 points on it.

The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an [[incidence structure]] {{math|(''P'', ''L'', ''I'')}} consisting of a set {{math|''P''}} of points, a set {{math|''L''}} of lines, and an [[incidence relation]] {{math|''I''}} that states which points lie on which lines.

The structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the [[Veblen–Young theorem]], there is no difference. However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the [[theorem of Desargues]] and are known as [[non-Desarguesian plane]]s. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically.<ref>{{harvnb|Baer|2005|p=71}}</ref>

It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. {{harvtxt|Coxeter|1969|p=231}} gives such an extension due to Bachmann.<ref>{{citation |first=F. |last=Bachmann |title=Aufbau der Geometrie aus dem Spiegelsbegriff |series=Grundlehren der mathematischen Wissenschaftern, 96 |publisher=Springer |place=Berlin |year=1959 |pages=76–77 }}</ref> To ensure that the dimension is at least two, replace the three point per line axiom above by:
* There exist four points, no three of which are collinear.
To avoid the non-Desarguesian planes, include [[Pappus's hexagon theorem|Pappus's theorem]] as an axiom;{{efn|As Pappus's theorem implies Desargues's theorem this eliminates the non-Desarguesian planes and also implies that the space is defined over a field (and not a division ring).}}
* If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear.
And, to ensure that the vector space is defined over a field that does not have even [[Characteristic (field)|characteristic]] include ''Fano's axiom'';{{efn|This restriction allows the real and complex fields to be used (zero characteristic) but removes the [[Fano plane]] and other planes that exhibit atypical behavior.}}
* The three diagonal points of a [[complete quadrangle]] are never collinear.

{{anchor|Projective subspace}}A '''subspace''' of the projective space is a subset {{math|''X''}}, such that any line containing two points of {{math|''X''}} is a subset of {{math|''X''}} (that is, completely contained in {{math|''X''}}).  The full space and the empty space are always subspaces.

The geometric dimension of the space is said to be {{math|''n''}} if that is the largest number for which there is a strictly ascending chain of subspaces of this form:
<math display="block">\varnothing = X_{-1}\subset X_{0}\subset \cdots X_{n}=P.</math>

A subspace {{math|''X''<sub>''i''</sub>}} in such a chain is said to have (geometric) dimension {{math|''i''}}. Subspaces of dimension 0 are called ''points'', those of dimension 1 are called ''lines'' and so on. If the full space has dimension {{math|''n''}} then any subspace of dimension {{nowrap|''n'' − 1}} is called a [[hyperplane]].

Projective spaces admit an equivalent formulation in terms of [[lattice (order)|lattice]] theory. There is a bijective correspondence between projective spaces and geomodular lattices, namely, [[subdirectly irreducible]], [[Compact element|compactly generated]], [[complemented lattice|complemented]], [[modular lattice]]s.<ref>Peter Crawley and [[Robert P. Dilworth]], 1973. ''Algebraic Theory of Lattices''. Prentice-Hall. {{isbn|978-0-13-022269-5}}, p. 109.</ref>

=== Classification ===
* Dimension 0 (no lines): The space is a single point.
* Dimension 1 (exactly one line): All points lie on the unique line.
* Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for {{math|1=''n'' = 2}} is equivalent to a [[projective plane]].  These are much harder to classify, as not all of them are isomorphic with a {{math|PG(''d'', ''K'')}}. The [[Desarguesian plane]]s (those that are isomorphic with a {{math|PG(2, ''K''))}} satisfy [[Desargues's theorem]] and are projective planes over division rings, but there are many [[non-Desarguesian plane]]s.
* Dimension at least 3: Two non-intersecting lines exist. {{harvtxt|Veblen|Young|1965}} proved the [[Veblen–Young theorem]], to the effect that every projective space of dimension {{math|''n'' ≥ 3}} is isomorphic with a {{math|PG(''n'', ''K'')}}, the {{math|''n''}}-dimensional projective space over some [[division ring]]&nbsp;{{math|''K''}}.

=== Finite projective spaces and planes ===
{{details|topic=finite projective planes|Projective plane#Finite projective planes}} 
[[Image:Fano plane.svg|thumb|right|The [[Fano plane]] ]]

A ''finite projective space'' is a projective space where {{math|''P''}} is a finite set of points. In any finite projective space, each line contains the same number of points and the ''order'' of the space is defined as one less than this common number. For finite projective spaces of dimension at least three, [[Wedderburn's little theorem|Wedderburn's theorem]] implies that the division ring over which the projective space is defined must be a [[finite field]], {{math|GF(''q'')}}, whose order (that is, number of elements) is {{math|''q''}} (a prime power). A finite projective space defined over such a finite field has {{math|''q'' + 1}} points on a line, so the two concepts of order coincide. Notationally, {{math|PG(''n'', GF(''q''))}} is usually written as {{math|PG(''n'', ''q'')}}.

All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are
{{block indent | em = 1.5 | text = 1, 1, 1, 1, 0, 1, 1, 4, 0, ... {{OEIS|id=A001231}}}}
finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the [[Bruck–Ryser–Chowla theorem|Bruck–Ryser theorem]].

The smallest projective plane is the [[Fano plane]], {{math|PG(2, 2)}} with 7 points and 7 lines. The smallest 3-dimensional projective space is [[PG(3,2)|{{math|PG(3, 2)}}]], with 15 points, 35 lines and 15 planes.

== Morphisms ==
Injective [[linear map]]s {{math|''T'' ∈ ''L''(''V'', ''W'')}} between two vector spaces {{math|''V''}} and {{math|''W''}} over the same field&nbsp;{{math|''K''}} induce mappings of the corresponding projective spaces {{math|'''P'''(''V'') → '''P'''(''W'')}} via:
{{block indent | em = 1.5 | text = {{math|[''v''] → [''T''(''v'')]}},}}
where {{math|''v''}} is a non-zero element of {{math|''V''}} and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces.  Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is [[well-defined]]. (If {{math|''T''}} is not injective, it has a [[null space]] larger than {{math|{{mset|0}}}}; in this case the meaning of the class of {{math|''T''(''v'')}} is problematic if {{math|''v''}} is non-zero and in the null space. In this case one obtains a so-called [[rational map]], see also ''[[Birational geometry]]''.)

Two linear maps {{math|''S''}} and {{math|''T''}} in {{math|''L''(''V'', ''W'')}} induce the same map between {{math|'''P'''(''V'')}} and {{math|'''P'''(''W'')}} [[if and only if]] they differ by a scalar multiple, that is if {{math|1=''T'' = ''λS''}} for some {{math|''λ'' ≠ 0}}.  Thus if one identifies the scalar multiples of the [[identity function|identity map]] with the underlying field&nbsp;{{math|''K''}}, the set of {{math|''K''}}-linear [[morphism]]s from {{math|'''P'''(''V'')}} to {{math|'''P'''(''W'')}} is simply {{math|1='''P'''(''L''(''V'', ''W''))}}.

The [[automorphism]]s {{math|'''P'''(''V'') → '''P'''(''V'')}} can be described more concretely. (We deal only with automorphisms preserving the base field&nbsp;{{math|''K''}}). Using the notion of [[ample line bundle|sheaves generated by global sections]], it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e., coming from a (linear) automorphism of the vector space {{math|''V''}}. The latter form the [[group (mathematics)|group]] [[general linear group|{{math|GL(''V'')}}]]. By identifying maps that differ by a scalar, one concludes that

{{block indent | em = 1.5 | text ={{math|1=Aut('''P'''(''V'')) = Aut(''V'') / ''K''<sup>×</sup> = GL(''V'') / ''K''<sup>×</sup> =: PGL(''V'')}},}}
the [[quotient group]] of {{math|GL(''V'')}} modulo the matrices that are scalar multiples of the identity. (These matrices form the [[center of a group|center]] of {{math|Aut(''V'')}}.) The groups {{math|PGL}} are called [[projective linear group]]s. The automorphisms of the complex projective line {{math|'''P'''<sup>1</sup>('''C''')}} are called [[Möbius transformation]]s.

== Dual projective space ==
When the construction above is applied to the [[dual space]] {{math|''V''<sup>∗</sup>}} rather than {{math|''V''}}, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of {{math|''V''}}.  That is, if {{math|''V''}} is {{math|''n''}}-dimensional, then {{math|'''P'''(''V''<sup>∗</sup>)}} is the [[Grassmannian]] of {{math|''n'' − 1}} planes in {{math|''V''}}.

In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles.  One would like to be able to associate a projective space to ''every'' quasi-coherent sheaf {{math|''E''}} over a scheme {{math|''Y''}}, not just the locally free ones.{{clarify|And how does the dual space help?|date=January 2011}} See [[Éléments de géométrie algébrique|EGA]]<sub>II</sub>, Chap. II, par. 4 for more details.

== Generalizations ==
; dimension : The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space {{math|''V''}} is generalized to [[Grassmannian manifold]], which is parametrizing higher-dimensional subspaces (of some fixed dimension) of {{math|''V''}}.
; sequence of subspaces : More generally [[flag manifold]] is the space of flags, i.e., chains of linear subspaces of {{math|''V''}}.
; other subvarieties : Even more generally, [[moduli space]]s parametrize objects such as [[elliptic curve]]s of a given kind.
; other rings : Generalizing to associative [[ring (mathematics)|ring]]s (rather than only fields) yields, for example, the [[projective line over a ring]].
; patching : Patching projective spaces together yields [[projective space bundles]].

[[Severi–Brauer variety|Severi–Brauer varieties]] are [[algebraic varieties]] over a field&nbsp;{{math|''K''}}, which become isomorphic to projective spaces after an extension of the base field&nbsp;{{math|''K''}}.

Another generalization of projective spaces are [[weighted projective space]]s; these are themselves special cases of [[toric variety|toric varieties]].<ref>{{ harvnb | Mukai | 2003 | loc=example 3.72 }}</ref>

== See also ==
* [[Geometric algebra]]

; Generalizations
* [[Grassmannian manifold]]
* [[Projective line over a ring]]
* [[Space (mathematics)]]

; Projective geometry
* [[projective transformation]]
* [[projective representation]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
* {{eom|id=P/p075350|first=V.V.|last= Afanas'ev|title=projective space}}
* {{citation|first=Reinhold|last=Baer|title=Linear Algebra and Projective Geometry|year=2005|orig-year=first published 1952| publisher=Dover|isbn=978-0-486-44565-6}}
* {{citation|first=Marcel|last=Berger|title=Geometry I|publisher=Springer-Verlag|year=2009|isbn=978-3-540-11658-5}}, translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation
* {{citation | last1=Beutelspacher | first1=Albrecht | last2=Rosenbaum | first2=Ute | title=Projective geometry: from foundations to applications | publisher=[[Cambridge University Press]] | isbn=978-0-521-48277-6 |mr=1629468 | year=1998}}
* {{citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Introduction to Geometry | publisher=John Wiley & Sons | location=New York | year=1974 | isbn=0-471-18283-4 | url-access=registration | url=https://archive.org/details/introductiontoge0002coxe }}
* {{citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link = Harold Scott MacDonald Coxeter | title=Projective geometry | publisher=University of Toronto Press | location=Toronto, Ont. | oclc=977732 |mr=0346652 | year=1969 | isbn=0-8020-2104-2}}
* {{citation | last1=Dembowski | first1=P. | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}
* Greenberg, M.J.; ''Euclidean and non-Euclidean geometries'', 2nd ed. Freeman (1980).
* {{citation | last1=Hartshorne | first1=Robin | author1-link = Robin Hartshorne | title=Algebraic Geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 |mr=0463157 | year=1977}}, esp. chapters I.2, I.7, II.5, and II.7
* Hilbert, D. and Cohn-Vossen, S.; ''[https://books.google.com/books?id=7WY5AAAAQBAJ&q=%22Projective+space%22 Geometry and the imagination]'', 2nd ed. Chelsea (1999).
* {{citation | last=Mukai | first=Shigeru | title=An Introduction to Invariants and Moduli | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | year=2003 | isbn=978-0-521-80906-1 }}
* {{citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=Projective geometry. Vols. 1, 2 | publisher=Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London |mr=0179666 | year=1965}} (Reprint of 1910 edition)

== External links ==
* {{MathWorld|urlname=ProjectiveSpace|title=Projective Space}}
* {{PlanetMath | urlname = ProjectiveSpace | title = Projective Space }}
* [https://web.archive.org/web/20060717030032/http://www.uwyo.edu/moorhouse/pub/planes/ Projective Planes of Small Order]

{{Dimension topics|state=uncollapsed}}

[[Category:Projective geometry]]
[[Category:Space (mathematics)]]