# Space (math)

A hierarchy of mathematical spaces: the scalar product induces a norm. The norm induces a metric. The metric induces a topology.

In mathematics, a room is a set of mathematical objects with a structure . This can be based on the underlying structures and / or on an additional mathematical structure . As a central example, a vector space consists of a set of objects, called vectors , that can be added or multiplied by a scalar (e.g. a number ) so that the result is again a vector of the same vector space and the associative and distributive laws apply. For example, real or complex numbers , number tuples , matrices or functions can serve as mathematical objects .

The term “space” in mathematics has changed significantly over time. While in classical mathematics space is understood to be the three-dimensional visual space , the geometric properties of which are completely defined by axioms , in modern mathematics spaces are merely abstract mathematical structures based on different concepts of the concept of dimension and whose properties are not fully defined by axioms become. The concept of space in physics has undergone a similar change since the 20th century .

Mathematical spaces can be classified on different levels, for example according to comparability, according to distinguishability and according to isomorphism . Rooms often form a hierarchy, that is, a room inherits all the properties of a higher-level room. For example, all scalar product spaces are also normalized spaces , since the scalar product induces a norm (the scalar product norm ) on the scalar product space .

Rooms are now used in almost all areas of mathematics, then as the busy linear algebra with vector spaces, the analysis with Folgen- and functional spaces , the geometry of affine and projective spaces, the topology with topological and uniform spaces, functional analysis with metric and normalized spaces, differential geometry with manifolds , measure theory with measurement and measure spaces and stochastics with probability spaces .

## history

### Before the golden age of geometry

In ancient mathematics , the term “space” was a geometric abstraction of the three-dimensional space observable in everyday life . Axiomatic methods have been an important tool in mathematical research since Euclid (around 300 BC) . Cartesian coordinates were introduced by René Descartes in 1637, thus establishing analytic geometry . At that time, geometric were doctrines as absolute truth regarded by intuition and logical thinking similar to the laws of nature could be recognized, and axioms were as obvious implications of definitions considered.

Two equivalence relations have been used between geometric figures : congruence and similarity . Translations , rotations and reflections map a figure into congruent figures and homotheties into similar figures. For example, all circles are similar to one another, but ellipses to circles are not. A third equivalence relation, which was introduced in 1795 in projective geometry by Gaspard Monge , corresponds to projective transformations . Under such transformations , not only ellipses, but also parabolas and hyperbolas can be mapped into circles; in the projective sense all of these figures are equivalent.

These relationships between Euclidean and projective geometry show that mathematical objects are not given along with their structure . Rather, every mathematical theory describes its objects in terms of some of their properties , precisely those which were formulated by axioms in the basis of the theory. Distances and angles are not mentioned in the axioms of projective geometry, so they cannot appear in their theorems. The question "what is the sum of the three angles of a triangle " only has a meaning in Euclidean geometry, but in projective geometry it is irrelevant.

A new situation arose in the 19th century: in some geometries the sum of the three angles of a triangle is well-defined, but different from the classical value (180 degrees ). In non-Euclidean hyperbolic geometry, which was introduced in 1829 by Nikolai Lobatschewski and in 1832 by János Bolyai (and, unpublished, in 1816 by Carl Friedrich Gauß ), this sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami and Felix Klein derived Euclidean models of hyperbolic geometry in 1868 and 1871, respectively , and thus justified this theory. A Euclidean model of a non-Euclidean geometry is a clever choice of objects in Euclidean space and relations between these objects that satisfy all axioms and thus all theorems of non-Euclidean geometry. The relationships of these selected objects of the Euclidean model mimick the non-Euclidean relationships. This shows that in mathematics the relationships between the objects, not the objects themselves, are of essential importance.

This discovery forced the abandonment of the claim of the absolute truth of Euclidean geometry. It showed that the axioms are neither obvious nor inferences from definitions; rather, they are hypotheses . The important physical question to what extent these correspond to experimental reality no longer has anything to do with mathematics. Even if a certain geometry does not correspond to experimental reality, its propositions still remain mathematical truths.

### The golden age and after

Nicolas Bourbaki calls the period between 1795 ( descriptive geometry by Monge) and 1872 ( Erlangen program by Klein) the "golden age of geometry". Analytical geometry had made great strides and was able to successfully replace theorems of classical geometry with calculations on invariants of transformation groups . Since then, new theorems of classical geometry have interested amateurs more than professional mathematicians. However, this does not mean that the legacy of classical geometry has been lost. According to Bourbaki, "classical geometry in its role as an autonomous and living science was overtaken and subsequently transformed into a universal language of contemporary mathematics".

Bernhard Riemann explained in his famous habilitation lecture in 1854 that every mathematical object that can be parameterized by real numbers can be viewed as a point in the -dimensional space of all such objects. Nowadays mathematicians routinely follow this idea and find it very suggestive to continue using the terminology of classical geometry almost everywhere. According to Hermann Hankel (1867), in order to fully appreciate the general validity of this approach, mathematics should be viewed as a “pure theory of forms, the purpose of which is not the combination of sizes or their images, numbers, but thought objects”. ${\ displaystyle n}$ ${\ displaystyle n}$

An object that is parameterized by complex numbers can be viewed as a point of a -dimensional complex space. However, the same object can also be parameterized by real numbers (the real and imaginary parts of the complex numbers) and can therefore be viewed as a point in -dimensional space. The complex dimension is therefore different from the real dimension. However, the concept of Dimension is much more complex. The algebraic concept of dimension relates to vector spaces , the topological concept of dimension to topological spaces . For metric spaces there is also the Hausdorff dimension , which can be non-integer especially for fractals . Function spaces are usually infinitely dimensional, as Riemann noted. Some rooms, for example dimensional rooms, do not allow any concept of a dimension at all. ${\ displaystyle n}$ ${\ displaystyle n}$${\ displaystyle 2n}$${\ displaystyle 2n}$

The space originally examined by Euclid is now called three-dimensional Euclidean space . Its axiomatization , begun by Euclid 23 centuries ago, was completed in the 20th century by David Hilbert , Alfred Tarski and George Birkhoff . Hilbert's system of axioms describes space using primitives that are not precisely defined (such as “point”, “between” or “congruent”), the properties of which are limited by a series of axioms. Such a definition from the ground up is of little use nowadays, as it does not show the relationship of this room to other rooms. The modern approach defines the three-dimensional Euclidean space rather algebraically via vector spaces and quadratic forms as affine space , the difference space of which is a three-dimensional scalar product space .

A room today consists of selected mathematical objects (for example functions between other rooms, sub- rooms of another room, or just the elements of a set) that are treated as points, as well as certain links between these points. This shows that spaces are just abstract mathematical structures.

## Systematics

### classification

Rooms can be classified on three levels. Since any mathematical theory defines its objects by just some of their properties, the first question that arises is: which properties?

The highest level of classification distinguishes rooms of different types. For example, Euclidean and projective spaces are of different types, since the distance between two points is defined in Euclidean spaces, but not in projective spaces. As a further example, the question “what is the sum of the three angles of a triangle” only makes sense in a Euclidean space, but not in a projective space. In non-Euclidean spaces this question makes sense and is just answered differently, which is not a distinction at the highest level. Furthermore, the distinction between a Euclidean plane and a three-dimensional Euclidean space is not a distinction at the highest level, since the question "what is the dimension" makes sense in both cases.

The second level of classification looks at answers to particularly important questions, among those that make sense at the highest level. For example, this level differentiates between Euclidean and non-Euclidean spaces, between finite-dimensional and infinite-dimensional spaces, between compact and non-compact spaces, etc.

The third level of classification looks at answers to all sorts of questions that make sense at the highest level. For example, this level differentiates between spaces of different dimensions, but not between a level of three-dimensional Euclidean space treated as two-dimensional Euclidean space and the set of all pairs of real numbers also treated as two-dimensional Euclidean space. Nor does it distinguish between different Euclidean models of the same non-Euclidean space. The third level classifies spaces in a more formal manner, except for isomorphism. An isomorphism between two spaces is a one-to-one correspondence between the points in the first space and the points in the second space that preserves all relationships between the points. Mutually isomorphic spaces are viewed as copies of the same space.

The concept of isomorphism sheds light on the highest level of classification. If there is a one-to-one correspondence between two spaces of the same type, then one can ask whether it is an isomorphism or not. This question makes no sense for rooms of different types. Isomorphisms of a space on itself are called automorphisms . Automorphisms of a Euclidean space are displacements and reflections. Euclidean space is homogeneous in the sense that any point in space can be transformed into any other point in space by a certain automorphism.

### Relationships between spaces

Topological terms (such as continuity , convergence, and open or closed sets ) are naturally defined in any Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the two topological spaces (called homeomorphism ), but the opposite direction is wrong: a homeomorphism can deform distances. According to Bourbaki, the structure of “topological space” is an underlying structure of “Euclidean space”.

The Euclidean axioms leave no degrees of freedom , they clearly determine all geometric properties of space. More precisely, all three-dimensional Euclidean spaces are mutually isomorphic. In this sense there is “the” three-dimensional Euclidean space. According to Bourbaki, the corresponding theory is univalent . In contrast, topological spaces are generally not isomorphic and their theory is multivalent . According to Bourbaki, the study of multivalent theories is the most important feature that distinguishes modern mathematics from classical mathematics.

## Important spaces

### Vector spaces and topological spaces

Vector spaces are algebraic in nature; there are real vector spaces (over the field of real numbers ), complex vector spaces (over the field of complex numbers ) and general vector spaces over any field. Every complex vector space is also a real vector space, so the latter space is based on the former, since every real number is also a complex number. Linear operations , which are given by definition in a vector space, lead to terms such as straight line (also plane and other sub-vector spaces ), parallel and ellipse (also ellipsoid ). However, orthogonal straight lines cannot be defined and circles cannot be singled out among the ellipses. The dimension of a vector space is defined as the maximum number of linearly independent vectors or, equivalently, the minimum number of vectors spanning the space ; it can be finite or infinite. Two vector spaces over the same body are isomorphic if and only if they have the same dimension.

Topological spaces are analytical in nature. Open sets , which are given by definition in topological spaces, lead to concepts such as continuity , path , limit value , interior , edge and exterior. However, terms such as uniform continuity , limitedness , Cauchy sequence or differentiability remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms ; they are one-to-one correspondence in both directions. The open interval is homeomorphic to the real number line , but not homeomorphic to the closed interval or to a circle. The surface of a cube is homeomorphic to a sphere , but not homeomorphic to a torus . Euclidean spaces of different dimensions are not homeomorphic, which is plausible but difficult to prove. ${\ displaystyle (0,1)}$ ${\ displaystyle (- \ infty, \ infty)}$ ${\ displaystyle [0,1]}$

The dimension of a topological space is not easy to define; the inductive dimension and the Lebesgue coverage dimension are used. Every subset of a topological space is itself a topological space (in contrast, only linear subspaces of a vector space are also vector spaces). Arbitrary topological spaces that are examined in the set- theoretical topology are too diverse for a complete classification, they are generally inhomogeneous. Compact topological spaces are an important class of topological spaces in which every continuous function is bounded . The closed interval and the extended real number line are compact; the open interval and the real number line are not. The geometric topology examined manifolds ; these are topological spaces that are locally homeomorphic to Euclidean spaces. Low-dimensional manifolds are fully classified except for homeomorphism. ${\ displaystyle [0,1]}$${\ displaystyle [- \ infty, \ infty]}$${\ displaystyle (0,1)}$${\ displaystyle (- \ infty, \ infty)}$

The structure of the topological vector space is based on the two structures vector space and topological space . That is, a topological vector space is both a real or complex vector space and a (even homogeneous) topological space. However, any combinations of these two structures are generally not topological vector spaces; the two structures must conform, that is, the linear operations must be continuous.

Every finite-dimensional real or complex vector space is a topological vector space in the sense that it has exactly one topology that makes it a topological vector space. The two structures “finite-dimensional real or complex vector space” and “finite-dimensional topological vector space” are therefore equivalent, i.e. they are mutually based. Correspondingly, every invertible linear transformation of a finite-dimensional topological vector space is a homeomorphism. In infinite dimension, however, various topologies conform to a given linear structure and invertible linear transformations are generally not homeomorphisms.

### Affine and projective spaces

It is convenient to introduce affine and projective spaces over vector spaces as follows. A -dimensional subspace of a -dimensional vector space is itself a -dimensional vector space and as such is not homogeneous: it contains a special point with the origin . By shifting a vector that is not in this sub-vector space, one obtains a -dimensional affine space that is homogeneous. In the words of John Baez , "an affine space is a vector space that has forgotten its origin". A straight line in an affine space is, by definition, its intersection with a two-dimensional linear subspace (a plane through the origin) of the -dimensional vector space. Every vector space is also an affine space. ${\ displaystyle n}$${\ displaystyle (n + 1)}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle (n + 1)}$

Every point of an affine space is its intersection with a one-dimensional sub-vector space (a straight line through the origin) of the -dimensional vector space. However, some one-dimensional subspaces are parallel to the affine space, in some ways they intersect at infinity . The set of all one-dimensional sub-vector spaces of a -dimensional vector space is, by definition, a -dimensional projective space. If one chooses a -dimensional affine space as before, then one observes that the affine space is embedded as a real subset in the projective space. However, the projective space itself is homogeneous. A straight line in the projective space corresponds, by definition, to a two-dimensional sub-vector space of the -dimensional vector space. ${\ displaystyle (n + 1)}$${\ displaystyle (n + 1)}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle (n + 1)}$

The affine and projective spaces defined in this way are algebraic in nature. They can be real, complex or generally defined over any field. Any real (or complex) affine or projective space is also a topological space. An affine space is a non-compact manifold, a projective space is a compact manifold.

### Metric and uniform spaces

Distances between points are defined in a metric space . Every metric space is also a topological space. Bounded sets and Cauchy sequences are defined in a metric space (but not directly in a topological space). Isomorphisms between metric spaces are called isometries . A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded in its completion. Every compact metric space is complete; the real number line is not compact but complete; the open interval is not complete. ${\ displaystyle (0,1)}$

A topological space is called metrizable if it is based on a metric space. All manifolds are metrizable. Every Euclidean space is also a complete metric space. In addition, all geometric terms that are essential for a Euclidean space can be defined using its metric. For example, the distance between two points and consists of all points , so the distance between and equals the sum of the distances between and and and . ${\ displaystyle A}$${\ displaystyle C}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle C}$

Uniform spaces do not allow spacing to be introduced, but nevertheless concepts such as uniform continuity, Cauchy sequences, completeness and completion can be defined. Every uniform space is also a topological space. Every topological vector space (regardless of whether it can be metrised or not) is also a uniform space. More generally, every commutative topological group is a uniform space. However, a non-commutative topological group has two uniform structures, a left-invariant and a right-invariant. Topological vector spaces are complete in finite dimensions, but generally not in infinite dimensions.

### Normalized spaces and scalar product spaces

The vectors in Euclidean space form a vector space, but each vector also has a length , in other words a norm . A real or complex vector space with norm is called normalized space. Every normalized space is both a topological vector space and a metric space. The set of vectors with norm less than one is called the unit sphere of normalized space. It is a convex and centrally symmetric set, but generally not an ellipsoid, for example it can also be a convex polyhedron . The parallelogram equation is generally not fulfilled in normalized spaces, but it applies to vectors in Euclidean spaces, which follows from the fact that the square of the Euclidean norm of a vector corresponds to the scalar product with itself. A Banach space is a completely normalized space. Many sequence or function spaces are infinite-dimensional Banach spaces.

A scalar product space is a real or complex vector space, which is equipped with a bilinear or sesquilinear form , which must meet certain conditions and is therefore called a scalar product . In a scalar product space , angles between vectors are also defined. Every dot product space is also a normalized space. A normalized space is based on a scalar product space if and only if the parallelogram equation is fulfilled in it or, equivalently, if its unit sphere is an ellipsoid. All -dimensional real dot product spaces are mutually isomorphic. One can say that the -dimensional Euclidean space is a -dimensional real scalar product space that has forgotten its origin. A Hilbert space is a complete scalar product space . Some sequence and function spaces are infinite-dimensional Hilbert spaces. Hilbert spaces are very important for quantum mechanics . ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$

### Differentiable and Riemannian manifolds

Differentiable manifolds are seldom called spaces, but they can be understood as such. Smooth (differentiable) functions, curves and maps , which are given by definition in a differentiable manifold, lead to tangent spaces . Every differentiable manifold is a (topological) manifold. Smooth surfaces in a finite-dimensional vector space, such as the surface of an ellipsoid, but not that of a polytope , are differentiable manifolds. Every differentiable manifold can be embedded in a finite-dimensional vector space. A smooth curve in a differentiable manifold has at every point a tangent vector that belongs to the tangent space at that point. The tangent space of a -dimensional differentiable manifold is a -dimensional vector space. A smooth function has a differential at every point , i.e. a linear functional on the tangent space. Real or complex finite-dimensional vector spaces, affine spaces, and projective spaces are all also differentiable manifolds. ${\ displaystyle n}$${\ displaystyle n}$

A Riemannian manifold or a Riemannian space is a differentiable manifold whose tangent space is equipped with a metric tensor . Euclidean spaces, smooth surfaces in Euclidean spaces, and hyperbolic non-Euclidean spaces are also Riemannian spaces. A curve in a Riemannian space has a length. A Riemann space is both a differentiable manifold and a metric space, where the distance corresponds to the length of the shortest curve. The angle between two curves that intersect at a point is the angle between their tangent vectors. If one waives the positivity of the scalar product on the tangential space, one obtains pseudo-Riemannian (and especially Lorentzian ) manifolds, which are important for the general theory of relativity .

### Measurement spaces, measurement spaces and probability spaces

If you do without distances and angles, but keep the volume of geometric bodies, you get into the field of dimension theory. In classical mathematics, a geometric body is much more regular than just a set of points. The edge of a geometric body has a volume of zero, so the volume of the body is equal to the volume of its interior, and the interior can be exploited by an infinite series of cubes. In contrast, the boundary of any set can have a non-zero volume, such as the set of all rational points within a given cube. Measure theory succeeded in extending the concept of volume (or any other measure) to an enormously large class of sets, the so-called measurable sets. In many cases, however, it is impossible to assign a measure to all quantities (see measure problem ). The measurable quantities form a σ-algebra . With the help of measurable quantities, measurable functions can be defined between measuring rooms.

In order to turn a topological space into a measuring space, one has to equip it with a σ-algebra. The σ-algebra of Borel sets is the most common, but not the only choice. Alternatively, a σ-algebra can be generated by a given family of sets or functions without taking any topology into account. Different topologies often lead to the same σ-algebra, such as the norm topology and weak topology on a separable Hilbert space. Each subset of a measuring room is itself a measuring room. Standard measuring rooms, also called standard Borel rooms, are particularly useful. Every Borel set, in particular every closed and every open set in a Euclidean space and more generally in a completely separable metric space (a so-called Polish space ) is a standard measurement space . All uncountable standard measuring spaces are isomorphic to one another.

A measurement space is a measurement space that is provided with a measurement . For example, a Euclidean space with the Lebesgue measure is a measure space. In integration theory , integratability and integrals of measurable functions are defined on dimensional spaces. Quantities of measure zero are called zero quantities . Zero sets and subsets of zero sets often appear as negligible exception sets in applications: For example, it is said that a property is valid almost everywhere if it is valid in the complement of a zero set. A measurement space in which all subsets of zero amounts can be measured is called complete .

A probability space is a measure space in which the measure of the whole space is equal to 1. In probability theory, the concepts of measure theory usually use their own designations, which are adapted to the description of random experiments: Measurable quantities are events and measurable functions between probability spaces are called random variables ; their integrals are expected values . The product of a finite or infinite family of probability spaces is again a probability space. In contrast to this, only the product of a finite number of spaces is defined for general dimensional spaces. Accordingly, there are numerous infinite-dimensional probability measures, for example the normal distribution , but no infinite-dimensional Lebesgue measure.

These spaces are less geometric. In particular, the idea of ​​dimension, as it is applicable in one form or another to all other spaces, cannot be applied to measurement spaces, measure spaces and probability spaces.

## literature

• Kiyosi Itô: Encyclopedic dictionary of mathematics . 2nd Edition. Mathematical society of Japan (original), MIT press (English translation), 1993 (English).
• Timothy Gowers, June Barrow-Green , Imre Leader: The Princeton Companion to Mathematics . Princeton University Press, 2008, ISBN 978-0-691-11880-2 (English).
• Nicolas Bourbaki: Elements of mathematics . Hermann (original), Addison-Wesley (English translation) - (French).
• Nicolas Bourbaki: Elements of the history of mathematics . Masson (original), Springer (English translation), 1994 (French).
• Nicolas Bourbaki: Elements of mathematics: Theory of sets . Hermann (original), Addison-Wesley (English translation), 1968 (French).
• Space . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

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9. ^ N. Bourbaki: Elements of the history of mathematics . 1994, p. 136 .
10. ^ A b N. Bourbaki: Elements of the history of mathematics . 1994, p. 138 .
11. ^ N. Bourbaki: Elements of the history of mathematics . 1994, p. 140 .
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13. ^ A b N. Bourbaki: Elements of mathematics: Theory of sets . 1968, Chapter IV.
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15. For example, the Gaussian number plane can be treated as a one-dimensional vector space and graded to a two-dimensional real vector space. In contrast to this, the real number line can be treated as a one-dimensional real vector space, but not as a one-dimensional complex vector space. See also body enlargement .