Projective space

Central projection of a railway line - the parallel rails seem to intersect at the vanishing point on the horizon.

The projective space is a term from the mathematical branch of geometry . This space can be understood as the set of all straight lines through the origin of a vector space . If the real two-dimensional vector space is , it is called a real projective line , and in the case it is called a real projective plane . Analogously, projective straight lines and projective planes over arbitrary bodies are defined as the sets of straight lines through the origin in a two- or three-dimensional vector space over the respective body. Projective planes can also be characterized axiomatically in the incidence geometry, and projective planes are also obtained that do not correspond to the straight lines in a vector space. ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle V=\mathbb {R} ^{3}}$

The idea of ​​projective spaces is related to the central projection from descriptive geometry and map design theory , or to the way in which the eye or a camera projects a three-dimensional scene onto a two-dimensional image. All points that are on a line with the lens of the camera are projected onto a common point. In this example, the underlying vector space is , the camera lens is the origin, and the projective space corresponds to the pixels. ${\displaystyle \mathbb {R} ^{3}}$

definition

The real-projective space is the set of all straight lines through the zero point in . Formally, it is defined as a set of equivalence classes as follows. ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle \mathbb {R} ^{n+1}}$

To be the equivalence relation ${\displaystyle \mathbb {R} ^{n+1}\setminus \{0\}}$

${\displaystyle x\sim y\Leftrightarrow \exists \lambda \in \mathbb {R} \setminus \{0\}\colon x=\lambda y}$

Are defined. In words this means that if and only if there is a such that it is equivalent to . All points on a straight line through the origin - the origin is not included - are therefore identified with one another and no longer differentiated. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \lambda \in \mathbb {R} \setminus \{0\}}$ ${\displaystyle x=\lambda y}$

The quotient space with the quotient topology is called real, -dimensional projective space and is also noted. ${\displaystyle \left(\mathbb {R} ^{n+1}\setminus \{0\}\right)/\sim }$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} P^{n}}$

In the case one speaks of the projective straight line (also: projective line ) and in the case of a projective plane . ${\displaystyle n=1}$ ${\displaystyle n=2}$

If one chooses instead of the complex vector space , one obtains with the analog definition with the complex projective space of the (complex) dimension as the space of the complex one-dimensional subspaces of the . ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle \mathbb {C} ^{n+1}}$ ${\displaystyle \lambda \in \mathbb {C} \setminus \{0\}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {C} ^{n+1}}$

The coordinates of the points of the projective space, which are equivalence classes of points , are noted by and are called homogeneous coordinates . (Correspondingly for the complex-projective space.) For the figure defines a bijection between and . ${\displaystyle (x_{0},\dotsc ,x_{n})\in \mathbb {R} ^{n+1}}$ ${\displaystyle [x_{0}:\ldots :x_{n}]\in \mathbb {R} P^{n}}$ ${\displaystyle n=1}$ ${\displaystyle [x_{0}:x_{1}]\rightarrow {\frac {x_{0}}{x_{1}}}}$ ${\displaystyle \mathbb {R} P^{1}}$ ${\displaystyle \mathbb {R} \cup \left\{\infty \right\}}$

More generally, projective spaces can also be constructed over any other bodies (instead of or ). ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

A more general term for projective space is used in synthetic geometry , especially in the case of the projective plane . The axiomatic of this more general term is presented in the main article Projective Geometry . ${\displaystyle n=2}$

Projective linear group (collineations)

→ Main article : Projective imaging

The projective linear group is the group of invertible projective mappings, it is defined as the quotient of below the equivalence relation ${\displaystyle \mathrm {PGL} (n+1,\mathbb {\mathbb {R} } )}$ ${\displaystyle \mathrm {GL} (n+1,\mathbb {\mathbb {R} } )}$

${\displaystyle A\sim B\Leftrightarrow \exists \lambda \in \mathbb {R} \setminus \{0\}\colon A=\lambda B}$.

The effect of on gives a well-defined effect of on . The images corresponding to the elements are projective, that is, collineations true to the double ratio here . In other words: ${\displaystyle \mathrm {GL} (n+1,\mathbb {R} )}$ ${\displaystyle \left(\mathbb {R} ^{n+1}\setminus \{0\}\right)}$ ${\displaystyle \mathrm {PGL} (n+1,\mathbb {R} )}$ ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle A\in \operatorname {PGL} (n+1,\mathbb {R} )}$ ${\displaystyle A:\mathbb {R} P^{n}\to \mathbb {R} P^{n}}$

1. They map the set of projective points bijectively onto themselves.
2. You map each straight line as a set of points onto a straight line (thus preserving the incidence structure).
3. The double ratio of any 4 points that lie on a straight line remains unchanged. This distinguishes projectivities from bijective, genuinely semilinear self-mapping of vector space.

An effect of on is defined analogously . ${\displaystyle \mathrm {PGL} (n+1,\mathbb {C} )}$ ${\displaystyle \mathbb {C} P^{n}}$

In the case of the projective straight acting on a pass-linear transformations. After the identification of with (or with ) takes effect or through . ${\displaystyle \mathrm {PGL} (2,\mathbb {R} )}$ ${\displaystyle \mathbb {R} P^{1}}$ ${\displaystyle \mathbb {R} P^{1}}$ ${\displaystyle \mathbb {R} \cup \{\infty \}}$ ${\displaystyle \mathbb {C} P^{1}}$ ${\displaystyle \mathbb {C} \cup \{\infty \}}$ ${\displaystyle \mathrm {PGL} (2,\mathbb {R} )}$ ${\displaystyle \mathrm {PGL} (2,\mathbb {C} )}$ ${\displaystyle \left({\begin{matrix}a&b\\c&d\end{matrix}}\right)z={\frac {az+b}{cz+d}}}$

Example: Riemann number ball

Stereographic back projections of the complex numbers and onto the dots and the Riemann sphere ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$

According to the above definition, the complex-projective straight line is the set of complex straight lines in which go through the origin . ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle (0,0)\in \mathbb {C} ^{2}}$

The complex projective straight line can also be called the real two-dimensional sphere or Riemann's number sphere

${\displaystyle S^{2}=\{(x,y,z)\in \mathbb {R} ^{3},x^{2}+y^{2}+z^{2}=1\}}$

grasp. The agreement with the above terms is as follows: Designate with the "North Pole". Look at the stereographic projection ${\displaystyle N:=(0,0,1)\in S^{2}}$

${\displaystyle f\colon S^{2}\setminus \{N\}\to \mathbb {R} ^{2}\cong \mathbb {C} }$,

which is given by . Clearly, a (real) straight line is laid through and the north pole and the point of intersection of this straight line with the equatorial plane is selected as the image point of the figure, with the north pole being identified. The correspondence between and in homogeneous coordinates is then . ${\displaystyle \textstyle (x,y,z)\mapsto \left({\frac {x}{1-z}},{\frac {y}{1-z}}\right)={\frac {x+iy}{1-z}}}$ ${\displaystyle (x,y,z)}$ ${\displaystyle \infty }$ ${\displaystyle S^{2}}$ ${\displaystyle \mathbb {C} P^{1}}$ ${\displaystyle (x,y,z)\mapsto \left[1:{\frac {x+iy}{1-z}}\right]=[1-z:x+iy]}$

properties

• The real and complex projective spaces are compact manifolds .
• The projective space is an example of a non-affine algebraic variety or a non-affine schema . In addition, the projective space has the structure of a toric variety . In the algebraic-geometric context, any body can be used instead of real or complex numbers .
• Sub- varieties of projective space are called projective varieties (also obsolete as projective manifolds).
• Locally homogeneous manifolds modeled locally according to the projective space are called projective manifolds .

topology

The projective straight line is homeomorphic to the circle . For the fundamental group of the projective space is the group Z / 2Z , the 2-fold superposition of the is the sphere . ${\displaystyle \mathbb {R} P^{1}}$ ${\displaystyle S^{1}}$ ${\displaystyle n>1}$ ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle \mathbb {S} ^{n}}$

For odd it is orientable , for even it is not orientable. ${\displaystyle n}$ ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle n}$

The projective plane is a non-orientable surface that cannot be embedded in the . But there are immersions of the in , for example the so-called Boyian surface . ${\displaystyle \mathbb {R} P^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} P^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$

The complex projective straight line is homeomorphic to the sphere , the quaternionic projective straight line is homeomorphic to , the Cayley projective straight line is homeomorphic to . ${\displaystyle \mathbb {C} P^{1}}$ ${\displaystyle S^{2}}$ ${\displaystyle \mathbb {H} P^{1}}$ ${\displaystyle S^{4}}$ ${\displaystyle CaP^{1}}$ ${\displaystyle S^{8}}$

All complex or quaternionic projective spaces are simply connected .

The Hopf fibers represent (for ) the unit sphere in on , the fiber is the unit sphere in . Fibers are obtained in this way ${\displaystyle \mathbb {K} =\mathbb {C} ,\mathbb {H} ,Ca}$ ${\displaystyle \mathbb {K} ^{2}}$ ${\displaystyle \mathbb {K} P^{1}}$ ${\displaystyle \mathbb {K} ^{1}}$

${\displaystyle S^{1}\to S^{3}\to S^{2},S^{3}\to S^{7}\to S^{4},S^{7}\to S^{15}\to S^{8}}$.

These fibers have Hopf invariant 1.

Projective sub-spaces and derived spaces

In this section, in the sense of the above general definition, a -dimensional projective space over any body is assumed, so the points of the space can be viewed as one-dimensional subspaces of . ${\displaystyle n}$ ${\displaystyle KP^{n}}$ ${\displaystyle K}$ ${\displaystyle K^{n+1}}$

• Each -dimensional subspace of is assigned a -dimensional projective subspace of . One also calls a (generalized, projective) plane , for hyperplane , for straight line in . The empty set is also regarded here as a projective subspace to which the null space of and is assigned as a dimension . ${\displaystyle k+1}$ ${\displaystyle (-1\leq k\leq n)}$ ${\displaystyle K^{n+1}}$ ${\displaystyle k}$ ${\displaystyle H}$ ${\displaystyle KP^{n}}$ ${\displaystyle H}$ ${\displaystyle k=n-1}$ ${\displaystyle k=1}$ ${\displaystyle KP^{n}}$ ${\displaystyle K^{n+1}}$ ${\displaystyle -1}$
• The intersection of two projective sub-spaces is in turn a projective sub-space.
• If one forms the linear envelope of their union set in to the subspaces, which are assigned to two projective spaces and , then a projective subspace, the connecting space (also noted as the sum ) of and , belongs to this subspace again . ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$ ${\displaystyle K^{n+1}}$ ${\displaystyle S_{1}\vee S_{2}}$ ${\displaystyle S_{1}+S_{2}}$ ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$
• The projective dimensional formula applies to the intersection and connection of projective sub-spaces :

${\displaystyle \operatorname {dim} (S_{1})+\operatorname {dim} (S_{2})=\operatorname {dim} (S_{1}\vee S_{2})+\operatorname {dim} (S_{1}\cap S_{2})}$.

• The set of all sub-spaces of the projective space forms a modular, complementary association of finite length with regard to the links “cut” and “connection” . ${\displaystyle {\mathcal {P}}^{n}}$ ${\displaystyle KP^{n}}$ ${\displaystyle \cap }$ ${\displaystyle \vee }$
• Each projective point can be assigned a homogeneous coordinate equation via its coordinates, the solution set of which describes a hyperplane. With the hyperplane coordinates defined in this way, the hyperplanes in turn form points of a projective space, the dual space (→ see projective coordinate system # coordinate equations and hyperplane coordinates ). ${\displaystyle KP^{n}}$ ${\displaystyle (KP^{n})^{D}}$
• More generally, the set of hyperplanes, which contain a fixed -dimensional subspace , forms a projective space, which is called a bundle , in a special case a tuft of hyperplanes. is called the carrier of the bundle or tuft. ${\displaystyle k}$ ${\displaystyle S}$ ${\displaystyle k=n-2}$ ${\displaystyle S}$

Axiomatic approach

See also : Hilbert's system of axioms of Euclidean geometry

When, in the second half of the 19th century, geometry was taken in a strictly axiomatic form and the axioms were then systematically varied, it made sense to replace the axiom of parallels by stipulating that two straight lines lying in one plane always intersect have to. However, this is incompatible with the axiom of arrangement II.3 .

If one restricts oneself to the incidence axioms , however, very simple and highly symmetrical axiom systems result, which also include the laws of known projective space.

Such a system of axioms, which only gets by with the basic terms “point”, “straight line” and “ incidence ”, is:

1. (Just Axiom) Are and two different points, so there is exactly one line , with and incised . ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle PQ}$ ${\displaystyle P}$ ${\displaystyle Q}$
2. ( Axiom of Veblen-Young ) are , , , four points, making and incident with a common point, so incise also and to a common point. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle AB}$ ${\displaystyle CD}$ ${\displaystyle AC}$ ${\displaystyle BD}$
3. (1st axiom of richness) Every straight line incurs with at least three points.
4. (2. Abundance axiom) There are at least two different straight lines.

An incidence structure that fulfills these axioms is then called a projective geometry .

The first axiom is a short version of the incidence axioms I.1 and I.2 .

The second axiom replaces the parallel axiom. If the term “plane” is appropriately defined within the framework of the other axioms, it just means that two straight lines of a plane always intersect. Replace it with the simpler (and stricter) axiom

2E. If and are two different straight lines, then there is exactly one point that incurs with and , ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$

so the corresponding structure is called a projective level .

The richness axioms 3 and 4 replace Hilbert's axiom I.8 . Structures that only fulfill axioms 1 to 3, but not 4, are called degenerate projective geometries . (Without exception, they are projective planes .)

The Fano level does not satisfy the Fano axiom !

Since both the arrangement axiom III.4 and the completeness axiom V.2 are missing, finite models for projective geometries are possible.

The simplest non-degenerate example is the Fano plane , which consists of seven points and seven straight lines; In the picture on the right, the "points" are the thick marked points, the "straight lines" are the lines and the circle.

A set of points of a projective space , which with two different points always also contains all points on their (according to axiom 1. unique) connecting line , is called a linear set . Linear sets play the role of the projective subspaces in projective geometry, so one also writes when is a linear set. ${\displaystyle \mathbb {P} }$ ${\displaystyle L\leq \mathbb {P} }$ ${\displaystyle L}$

• The simplest (though not the smallest) type of linear set is a series of points, i.e. the set of points on a straight line.
• An arbitrary set of points in space produces a well-defined minimal linear set ${\displaystyle M}$

${\displaystyle \langle M\rangle :=\bigcap _{M\subseteq L\leq \mathbb {P} }L,\quad }$the intersection of all linear sets in which is contained as a subset. ${\displaystyle M}$

• If and for every point , then a minimal generating system or a point basis is called of . The number of elements of such a point base is independent of the choice of point base. The number is called the projective dimension of , it can be a natural number or more generally an infinite cardinal number , in the latter case the linear set is often only called infinitely dimensional . ${\displaystyle \langle M\rangle =L}$ ${\displaystyle B\in M:\;\langle M\setminus \{B\}\rangle \neq L}$ ${\displaystyle M}$ ${\displaystyle L}$ ${\displaystyle m}$ ${\displaystyle L}$ ${\displaystyle d=m-1}$ ${\displaystyle L}$ ${\displaystyle -1,0}$

Examples

1. According to the definition mentioned, the empty set is itself a linear set: it contains the points of all necessary connecting lines, namely none. Your dimension is . ${\displaystyle d=\#\emptyset -1=-1}$
2. Likewise, every single-point set is a linear set, so its dimension is always . ${\displaystyle 0}$
3. Every row of points is a one-dimensional linear set, because it is generated by any two different points on the carrier line.

These three types of linear sets meet (together with the at most one straight line that goes through two different points of the linear set and the incidence restricted to this partial structure) the first three incidence axioms (more or less trivial) but not the fourth are degenerate projective spaces . A linear set that contains three points that do not lie on a common straight line also satisfies the fourth axiom of incidence and is therefore itself a projective space. The dimension of this linear set is then at least 2. Please note that the term plane in the description above is to be understood axiomatically and is not directly related to the dimension term for linear sets. Degenerate projective planes , which are linear quantities in a projective space, always belong to one of the three types mentioned above and therefore have a projective dimension as linear quantities . The total space is of course also a linear quantity and accordingly has a well-defined dimension. ${\displaystyle d\in \{-1,0,1\}}$

Additional axioms

Closure properties

As additional axioms, two classical inference theorems , the Desargues theorem and the Pappos theorem, are particularly important: These axioms are each equivalent to the fact that the geometry can be coordinated via a class of ternary bodies determined by the axioms :

• Exactly when Desargues' theorem applies to every two-dimensional linear set of space, space can be coordinated by a sloping body . These conditions are always met for at least three-dimensional spaces. This last statement is a sentence by David Hilbert .
• Exactly when Pappos's theorem applies to every two-dimensional linear set of space, space can be coordinated by a commutative body . These conditions are not always met even for three- and higher-dimensional ones .

The inference theorems were (implicitly) proved as theorems that apply in real two- or three-dimensional geometry by the mathematicians after whom they are named. Implicit because in their time there was neither an axiomatic description of the modern algebraic concept of the body nor even of the field of real numbers . A modern "no-closure axiom" is the Fano axiom . It is of great importance when studying quadrics . For these investigations one usually has to demand Pappos' axiom. If the Fano axiom also applies, then the coordinate body of the space does not have the characteristic , that is, a quadratic equation "mostly" has no or two solutions and one can, for example, distinguish between tangents and non- tangents in a conic section . ${\displaystyle 2}$

Order properties and topological properties

A projective space is arranged if a separating relationship is defined on each straight line in such a way that this relationship is retained for any projectivities . The separating relationship continues the above-described Hilbert affine arrangement projectively: If a point is affine between the points , then the pair of points separates the point from the far point of the (projectively closed) straight line . The interrelation on the affine line satisfies Pasch's axiom . If the product topology is formed from the order topology on any straight line ( is the dimension of the affine space), then this is a "compatible" topology for the space on the basis of Pasch's axiom: The affinities of the space are continuous with regard to this topology . ${\displaystyle B\in g}$ ${\displaystyle A,C\in g}$ ${\displaystyle A,C}$ ${\displaystyle B}$ ${\displaystyle H_{\infty }\cap {\bar {g}}}$ ${\displaystyle {\bar {g}}}$ ${\displaystyle g}$ ${\displaystyle g^{d}}$ ${\displaystyle d\geq 2}$

This topology can now be continued (initially on individual straight lines) by making the affine sets of intermediate points ("open intervals") the basis of a topology on the projective straight line with any choice of the far point and providing the space with the corresponding product topology. Thus a projective plane becomes a topological projective plane and a higher-dimensional space (more precisely: the set of its points) becomes a topological space in which the projectivities are homeomorphisms . ${\displaystyle {\bar {g}}}$

Such an arrangement of the affine and projective spaces is only possible (necessary condition) if the following applies in a coordinate ternary body : If this "sum" is bracketed with more than one summand (in the ternary body, the associative law does not have to apply for the addition, is one Loop ) then is . From this it follows for every arranged space: it and its coordinate range is infinite. If the space is also Desargue's, i.e. if it fulfills Desargue's closure axiom, then its coordinate oblique body has the characteristic 0. ${\displaystyle (K,T,0,1)}$ ${\displaystyle a+a+\cdots +a=0}$ ${\displaystyle (K,+,0)}$ ${\displaystyle a=0}$

More generally, one can also define a topology axiomatically in a topological space; this is shown for the two-dimensional case in the article Topological projective plane . Each projective space allows at least one topology in the sense of the requirements presented there, namely the discrete topology . This is usually not an "interesting" topology.

On projective spaces over oblique bodies or bodies such as the body of complex numbers and the real quaternion oblique body , which are finite-dimensional vector spaces over an arranged lower body (in the examples ), one can use the affine section (more precisely: in the group of projective perspectives with a fixed Fixed point hyperplane and any centers on this hyperplane) introduce a topology: This group, the affine translation group, is a (left) vector space above and thus also above , which means that the order topology, which originates from the arrangement of the straight lines, can also be applied to the affine and projective space transferred over . ${\displaystyle S}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle H_{\infty }}$ ${\displaystyle S}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle S}$

properties

In the following we understand by a projective space a structure of points and straight lines with an incidence relation which fulfills the axioms of Veblen-Young mentioned above and in which there are two straight lines that are foreign to the point; the projective planes are therefore excluded. Then the following theorems apply:

Desargues' theorem applies to every projective space of dimension : If there are different points such that , and determine three different straight lines, then the three points of intersection of with , with and with lie on a straight line. With the help of this theorem it can be shown: Every projective space can be described by homogeneous coordinates in a left vector space over a sloping body . The left vector space is at least four-dimensional, but its dimension can also be any infinite cardinal number . The inclined body is commutative, i.e. a body if and only if the Pappos theorem (-Pascal) applies in the geometry of this space . This is always the case in finite Desarguese planes (because finite skew bodies are necessarily commutative according to Wedderburn's theorem ). ${\displaystyle \geq 3}$ ${\displaystyle O,A,B,C,A',B',C'}$ ${\displaystyle O,A,A'}$ ${\displaystyle O,B,B'}$ ${\displaystyle O,C,C'}$ ${\displaystyle AB}$ ${\displaystyle A'B'}$ ${\displaystyle BC}$ ${\displaystyle B'C'}$ ${\displaystyle CA}$ ${\displaystyle C'A'}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K}$

In synthetic geometry, the “non-Desargues” planes, in which Desargues's theorem does not apply, are of particular interest, especially the finite ones. The order of a finite projective plane is the number of points on one, i.e. each, straight line, reduced by 1. It is an unproven conjecture that any finite projective plane is of prime power order (like the Desarguean planes). A sentence by Bruck and Ryser excludes many orders. He says: if or is the order of a projective plane, then is the sum of two square numbers. The following numbers are therefore not orders of projective planes: ${\displaystyle n=4k+1}$ ${\displaystyle n=4k+2}$ ${\displaystyle n}$ ${\displaystyle 6,14,21,22,30,33,38,42,46,\ldots }$

With great use of computers it was shown that no projective level of order exists. The smallest orders, for which the question of existence or non-existence is unsolved, are The smallest order of a non-Desarguean projective plane , compare the section Examples of order 9 in the article Ternary bodies . ${\displaystyle 10}$ ${\displaystyle 12,15,18,20.}$ ${\displaystyle 9}$

literature

1. ↑ ^{a b c d e f} Beutelspacher (1982)
2. ↑ ^{a b} David Hilbert: Fundamentals of Geometry . 14th edition. Teubner, Stuttgart 1999, ISBN 3-519-00237-X ( archive.org - first edition: 1899).
3. ↑ Historically, it should also be noted that, contrary to the implication: “The Desargues theorem follows from the Pappos theorem”, the Hessenberg theorem does not follow trivially from the fact that every oblique body is a body: Only the Desarguesian sentence is suitable (based on current knowledge) for the introduction of coordinates. Therefore, the validity of Hessenberg's theorem must be proven in any projective space without coordinates .

• D. Hilbert, S. Cohn-Vossen: Illustrative geometry. With an appendix: The simplest basic concepts of topology by Paul Alexandroff. Reprint of the 1932 edition. Scientific Book Society, Darmstadt, 1973
• W. Massey: Algebraic topology: An introduction. Harcourt, Brace & World, Inc., New York 1967.
• R. Hartshorne: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. ISBN 0-387-90244-9

Web links

• Projective Space in the Encyclopaedia of Mathematics
• Algebra and Geometry through Projective Spaces