Space (mathematics)
A room is to be a set of mathematical objects with an additional mathematical structure. As a key example, there is a vector space from a set of objects, called vectors (eg a number) can be multiplied so that the result is again a vector of the same vector space and the associative and distributive laws are added to or by a scalar. As mathematical objects can serve, for example, real or complex numbers, Zahlentupel, matrices or functions.
The term "space" has changed significantly in mathematics over time. While it is understood in the classical mathematics under the three-dimensional space perception space whose geometrical properties are completely defined by axioms, rooms are in modern mathematics merely abstract mathematical structures with different concepts of dimension, where only some properties are defined by axioms. A similar change has also witnessed the concept of physical space since the 20th century.
Mathematical spaces can be classified on different levels, including comparability, after differentiation and by isomorphism. Spaces often form a hierarchy, that is, a space inherits all properties of its parent space. For example, all Skalarprodukträume also normed spaces, since the scalar product a standard ( the Skalarproduktnorm ) induced on the inner product.
Rooms are now used in almost all areas of mathematics, then employs approximately linear algebra with vector spaces that Analysis with consequences and function rooms, the geometry of affine and projective spaces, the topology with topological and uniform spaces, Functional Analysis with metric and normed spaces, differential geometry with manifolds, measure theory with measurement and measure spaces and the stochastic with probability spaces.
- 2.1 classification
- 2.2 Relations between spaces
- 3.1 Vector spaces and topological spaces
- 3.2 Affine and projective spaces
- 3.3 Metric and uniform spaces
- 3.4 Normed spaces and Skalarprodukträume
- 3.5 Differentiable and Riemannian manifolds
- 3.6 Measuring spaces, measure spaces and probability spaces
History
Before the golden age of geometry
In mathematics of antiquity, the term "space" was a geometrical abstraction of the observable in daily life three-dimensional space. Since Euclid ( about 300 BC) are axiomatic methods is an important tool in mathematical research. By René Descartes in 1637 Cartesian coordinates were introduced and thus founded analytic geometry. At this time, geometric theorems were considered as absolute truth, which is similar to the laws of nature could be discerned by intuition and logical thinking, and axioms were considered obvious consequences of the definitions.
Between geometric figures two equivalence relations were used: congruence and similarity. Translations, rotations and reflections form a figure into congruent figures from homotheties and in similar figures. However, for example, all circles are similar to each other, ellipses into circles not. A third equivalence relation, which was introduced in 1795 by Gaspard Monge projective geometry corresponds to projective transformations. Under such transformations not only ellipses, parabolas and hyperbolas but also can be mapped into circles; in the projective sense, all these figures are equivalent.
These relations between the Euclidean and projective geometry show that mathematical objects are not placed together with their structure. Rather, any mathematical theory describes its objects by some of their properties, precisely those that have been formulated by axioms at the basis of the theory. Distances and angles are not mentioned in the axioms of projective geometry, so they may not appear in their records. The question " what is the sum of the three angles of a triangle " has a meaning only in Euclidean geometry, but in projective geometry it is pointless.
In the 19th century a new situation occurred: in some geometries is the sum of the three angles of a triangle is well-defined, but different from the classical value ( 180 degrees). In the non-Euclidean hyperbolic geometry, the 1829 Nikolai Lobachevsky and 1832 by János Bolyai ( and, unpublished, 1816 by Carl Friedrich Gauss ) was introduced, this sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami and Felix Klein introduced her in 1868 or 1871 Euclidean models of hyperbolic geometry, and thus justify 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 sets of non-Euclidean geometry. The relationships of these selected objects of the Euclidean model mimic the non-Euclidean relations. This shows that in mathematics, the relations between the objects, not the objects themselves, is of essential importance.
This discovery forced the departure from the claim of absolute truth of Euclidean geometry. It showed that the axioms, nor are obvious consequences of definitions; rather, they are hypotheses. The important physical question of to what extent they comply with the experimental reality, has nothing to do with mathematics. Even if a particular geometry does not match the experimental reality, their rates still remain mathematical truths.
The golden age and then
Nicolas Bourbaki calls the period between 1795 (descriptive geometry of Monge ) and 1872 ( Erlanger program of Klein) the "golden age of Geometry". The analytic geometry had made great progress and was able to successfully replace sets of classical geometry by invoices invariants of transformation groups. Since that time, new sets of classical geometry more interested amateurs and professional mathematicians. This does not mean that the legacy of classical geometry was lost. After Bourbaki " classical geometry overtaken in their role as autonomous and vibrant science and subsequently converted into a universal language of contemporary mathematics " was.
Bernhard Riemann stated in 1854 in his famous habilitation lecture that every mathematical object that can be parameterized by real numbers, as a point in the -dimensional space of all such objects can be viewed. Today, mathematicians routinely follow this idea and find it very suggestive, almost re-using the terminology of classical geometry everywhere. After Hermann Hankel (1867 ) should, in order to fully appreciate the generality of the approach, mathematics as "pure theory of forms whose purpose is not the combination of variables or their images, the numbers, but objects of thought " view.
An object that is parameterized by complex numbers can be regarded as a point - dimensional complex space. However, the same object can also be real numbers ( the real and imaginary parts of complex numbers ) are parameterized and are therefore considered as a point in -dimensional space. The complex dimension that is different from the real dimension. The concept of dimension is, however, considerably more complex. The algebraic concept of dimension refers to vector spaces, the topological concept of dimension for topological spaces. For metric spaces, there is also the Hausdorff dimension for fractals can be non- integer to be special. Function spaces are usually infinite dimensional, as Riemann noted. Some areas, like the measure spaces, ever allow no concept of a dimension.
The originally studied by Euclid space is called three-dimensional Euclidean space today. His axiomatization of Euclid started 23 centuries ago, was completed by David Hilbert, Alfred Tarski and George Birkhoff in the 20th century. Hilbert's axiom system describes the space above is not precisely defined primitives ( such as " point ", " between " or " congruent "), whose properties are limited by a number of axioms. Such a definition from the ground up is now of little use, since it does not shows the relation of this room to other rooms. The modern approach defines the three-dimensional Euclidean space rather algebraic over vector spaces and quadratic forms as affine space whose difference is a three-dimensional inner product space.
An area today consists of selected mathematical objects ( for example, functions between other rooms, part rooms of another space, or even the elements of a set ), which are treated as points, and from certain linkages between these points. This shows that only spaces are abstract mathematical structures. One might expect that the geometric areas are different mathematical structures, but this is not always the case. For example, a differentiable manifold is much more geometric than a measure space, but no one would describe as " differentiable space ".
System
Classification
Space can be classified in three levels. After each mathematical theory defines its objects only by some of their properties, is the first question that arises: what characteristics?
The highest level of classification distinguishes rooms of different types. For example, Euclidean and projective spaces of different types, as the distance between two points is defined in Euclidean spaces, not in projective spaces, however. As another example, the question arises " what is the sum of the three angles of a triangle " only in a Euclidean space, but not in a projective space sense. In non-Euclidean spaces this question makes sense and will only be answered differently, which is not a distinction at the highest level. Furthermore, the distinction between a Euclidean plane and a three-dimensional Euclidean space, no distinction at the highest level since the question "what is the dimension " in both cases makes sense.
At the second level of classification is considered answers to the most important issues, including those that would result in the highest level sense. For example, distinguishes between Euclidean and Non-Euclidean plane these spaces, between finite and infinite-dimensional spaces, between compact and noncompact spaces, etc.
At the third level of classification are answers to all possible questions that make sense at the highest level is considered. For example, this level differs between spaces of different dimensions, but not between a plane of the three-dimensional Euclidean space treated also treated as two-dimensional Euclidean space as a two dimensional Euclidean space and the set of all pairs of real numbers. Likewise, it does not distinguish between different models of the same Euclidean non-Euclidean space. Formal classified the third level rooms up to isomorphism. An isomorphism between two rooms is a one-to -one correspondence between the points of the first space and the points of the second space, which receives all the relationships between the points. Mutually isomorphic spaces are considered to copies of the same space.
The concept of isomorphism sheds light on the highest level of classification. Is a one-to -one correspondence between two rooms of the same type given, then one can ask whether there is an isomorphism or not. This question does not make sense for rooms of different types. Isomorphisms of a space to itself are called automorphisms. Automorphisms of a Euclidean space are shifts and reflections. Euclidean space is homogeneous in the sense that each point of the space may be transformed by a specific automorphism to any other point in space.
Relationships between spaces
Topological concepts (like continuity, convergence, and open or closed sets ) are defined in a natural way in any Euclidean space. In other words, each Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the two topological spaces ( called homeomorphism ), the reverse direction is wrong: a homeomorphism can deform distances. After Bourbaki is the structure of " topological space " is an underlying structure of " Euclidean space ".
The Euclidean axioms can be no degrees of freedom, they unambiguously determine all the geometric properties of space. Specifically, all three-dimensional Euclidean spaces are mutually isomorphic. In this sense, there is "the " three-dimensional Euclidean space. Following Bourbaki, the corresponding theory is univalent. In contrast, topological spaces are generally not isomorphic and their theory is multivalent. After Bourbaki studying multivalent theories is the most important feature that distinguishes the modern mathematics of classical mathematics.
Important rooms
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 an arbitrary field. Each complex vector space is a real vector space, the latter space is therefore based on the former, since every real number is a complex number. Linear operations which are given in a vector space, by definition, lead to terms such as straight line ( also level and other subspaces ), Parallel and ellipse (also ellipsoid). However Orthogonal lines can not be defined and circles can not be singled out among the ellipses. The dimension of a vector space is defined as the maximum number of linearly independent vectors, or equivalently, as the minimum number of vectors that span the space defined; it can be finite or infinite. Two vector spaces over the same body are isomorphic if they have the same dimension.
Topological spaces are analytical in nature. Open sets, which are given in topological spaces, by definition, lead to terms such as continuity, path, limit, home affairs, border and exterior. However, terms such as uniform continuity, boundedness, differentiability or Cauchy sequence 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 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.
The dimension of a topological space is not easy to define; the inductive dimension and the dimension Lebesgue'sche coverage it used to be. Every subset of a topological space is itself a topological space ( in contrast, only linear subspaces of a vector space and vector spaces). Any topological spaces, which are investigated in the set-theoretic topology, are too diverse for a complete classification, they are not homogeneous in general. Compact topological spaces are an important class of topological spaces in which every continuous function is limited. The closed interval and the extended real line are compact; the open interval and the real line are not. The geometric topology of manifolds is investigated; these are topological spaces which are locally homeomorphic to Euclidean spaces. Low-dimensional manifolds are completely classified up to homeomorphism.
The two structures vector space and topological space underlying the structure of the topological vector space. That is a topological vector space is both a real or complex vector space, as well as an ( even homogeneous ) topological space. However, any combination of these two structures are generally not topological vector spaces; the two structures must be compliant, that is, the linear operations must be continuous.
Every finite real or complex vector space is a topological vector space in the sense that he just carries a 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, that are themselves based on each other. Accordingly, any invertible linear transformation of a finite-dimensional topological vector space is a homeomorphism. In infinite dimension, however, different topologies are conforming to a 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. One- dimensional subspace of a - dimensional vector space is itself an - dimensional vector space and as such is not homogeneous: it contains the origin of a particular point. By shifting to a not lying in this subspace vector to obtain a - dimensional affine space which is homogeneous. In the words of John Baez is " an affine space is a vector space, which has forgotten its origin ." Is a straight line in an affine space, by definition, their section with a two-dimensional linear subspace ( a plane passing through the origin) of the - dimensional vector space. Each vector space is an affine space.
Each point of an affine space is to be cut with a one dimensional Unterverktorraum ( a line through the origin ) of the - dimensional vector space. However, some one-dimensional subspaces are parallel to the affine space, in a way, they meet at infinity. The set of all one-dimensional subspaces of a - dimensional vector space is, by definition, one - dimensional projective space. If you take one - dimensional affine space as before, one observes that the affine space is embedded as a proper subset in the projective space. The projective space itself, however, is homogeneous. A straight line corresponds to the projective space, by definition, a two-dimensional subspace of the - dimensional vector space.
Values defined in this way, affine and projective spaces are of algebraic nature. You can be real, complex or broadly defined over an arbitrary field. Every 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
In a metric space distances between points to be defined. Every metric space is a topological space. In a metric space (but not directly in a topological space ) Limited quantities and Cauchy sequences are defined. Isomorphism between metric spaces are called isometries. A metric space is called complete if all Cauchy sequences converge. Any non- complete space is isometrically embedded in its completion. Every compact metric space is complete; the real line is not compact but complete; the open interval is not complete.
A topological space is called metrizable if it is a metric space based. All manifolds are metrizable. Every Euclidean space is a complete metric space. In addition, all the geometric items that are essential for the Euclidean space are defined by its metric. For example, the distance between two points and consists of all points such that the distance between and is equal to the sum of the distances between and and and.
Uniform spaces allow not introduce distances, but still to define terms such as uniform continuity, Cauchy sequences, completeness and completion. Each uniform space is a topological space. Every topological vector space (whether or not metrizable ) is also a uniform space. General any commutative topological group is a uniform space. However, a non-commutative topological group carries two uniform structures, a left -invariant and right -invariant. Topological vector spaces are finite dimensions fully, in infinite dimensions but generally not.
Normed spaces and Skalarprodukträume
The vectors in a Euclidean space form a vector space, but each vector has a length, in other words, a norm. A real or complex vector space with a norm is called normed space. Every normed space is both a topological vector space and a metric space. The set of vectors with norm smaller than one is called a unit ball of the normed space. It is a convex and centrally symmetric amount, generally but not ellipsoid, for example, it may also be a convex polyhedron. The parallelogram law is not fulfilled in normed spaces in general, but it applies to vectors in Euclidean spaces, which it follows that the square of the Euclidean norm of a vector corresponds to the scalar product with itself. A Banach space is a complete normed space. Many episodes or functional spaces are infinite-dimensional Banach spaces.
An inner product is a real or complex vector space equipped with a bilinear or sesquilinear form, must meet certain conditions and is therefore called the scalar product. In an inner product and angle between vectors are defined. Each scalar product space is also a normed space. A normed space is an inner product if and only applicable if the parallelogram law is fulfilled in him or, equivalently, if its unit ball is an ellipsoid. All - dimensional real Skalarprodukträume are mutually isomorphic. One can say that the -dimensional Euclidean space is a - dimensional real inner product, which has forgotten its origin. A Hilbert space is a complete inner product. Some consequences and function spaces are infinite-dimensional Hilbert spaces. Hilbert spaces are very important for quantum mechanics.
Differentiable and Riemannian manifolds
Differentiable manifolds are rarely referred to as spaces, but can be understood as such. Smooth ( differentiable ) functions, graphs and maps, which are given in a differentiable manifold, by definition, lead to tangent. Every differentiable manifold is a ( topological ) manifold. Smooth surfaces in a finite dimensional vector space, such as the surface of an ellipsoid, but not a polytope, are differentiable manifolds. Every differentiable manifold can be embedded into a finite-dimensional vector space. A smooth curve in a differentiable manifold has a tangent which is part of the tangent at this point at each point. The tangent of a -dimensional differentiable manifold is an - dimensional vector space. A smooth function has at each point a differential, ie, a linear functional on the tangent space. Real or complex finite dimensional vector spaces, affine spaces and projective spaces are all also differentiable manifolds.
A Riemannian manifold or 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, the distance of the length of the shortest curve. The angle between the two curves intersect at a point that is the angle between its tangent. When you remove the positivity of the scalar product on the tangent space to obtain pseudo- Riemannian (and in particular Lorentz ) manifolds, which are important for the general theory of relativity.
Measurable spaces, measure spaces and probability spaces
When you remove distances and angles, but keeps the volume of geometric bodies in, you get into the field of measure theory. A geometric body is in classical mathematics much more regular than just a set of points. The edge of a geometric body has the volume to zero, therefore, the volume of the body is equal to the volume of the interior thereof, and the inside can be exhausted by means of an infinite sequence of dice. In contrast, the edge of any quantity may have a volume equal to zero, such as the set of rational points within a given cube. The measure theory it (or other measure ) extend the notion of the volume of an enormously large class of quantities, called measurable quantities succeeded. In many cases, however, it is impossible to allocate all amounts a measure (see measurement problem ). The measurable sets form a σ - algebra case. With help of measurable quantities to measurable functions between measurement areas can be defined.
To make a topological space in a measurement space, you have to equip it with a σ - algebra. The σ - algebra of Borel sets is the most common but not the only choice. Alternatively, can also be generated by a given family of sets or functions, a σ - algebra, without any topology to take into consideration. Frequently, different topologies lead to the same σ - algebra, such as the norm topology and weak topology on a separable Hilbert space. Every subset of a measurement space is itself a measurable space. Standard measuring rooms, also called standard Borel spaces, are particularly useful. Every Borel set, so in particular each completed and each open set in a Euclidean space, and more generally in a complete separable metric space (called a Polish space ) is a standard measurement space. All uncountable standard measurable spaces are isomorphic to each other.
A measure space is a measurable space, which is provided with a measure. A Euclidean space with the Lebesgue measure, for example, be a measure space. In the theory of integration of integration and integrals of measurable functions on measure spaces are defined. Sets of measure zero are called null sets. Zero sets and subsets of null sets occur in applications often than negligible quantities except in appearance: One speaks about the fact that a property holds almost everywhere if it applies in the complement of a null set. A dimensional space in which all the subsets of zero measurable quantities is, completely.
A probability space is a measure space, wherein the amount of the whole area is equal to 1. In probability theory own designations are used for the measure theoretic terms usually used are adapted to the description of random experiments: Measurable quantities are measurable events and functions between probability spaces are called random variable; their integrals are expected values . The product of a finite or infinite family of probability spaces is a probability space again. In contrast, only the product of finitely many spaces is defined for general measure spaces. Accordingly, there are a number of infinite- probability measures, such as the normal distribution, but no unendlichdimensionales Lebesgue measure.
These spaces are less geometrically. In particular, can the idea of the dimension, as it is applicable to all the other rooms in one form or another, do not use on measurable spaces, measure spaces and probability spaces.