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## AMS :: Bull. Amer. Math. Soc. -- Volume 83, Number 2

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The True furnace career 's 16 Process. In other words, the latter transition is injective one-to-one , while the former transition is not injective many-to-one. Both transitions are not surjective , that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special not general case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology for instance, it must be non-compact, and connected, etc. See for example Fig. Such topology is non-unique in general, but unique when the real linear space is finite-dimensional.

For these spaces the transition is both injective and surjective, that is, bijective ; see the arrow from "finite-dim real linear topological" to "finite-dim real linear" on Fig. The inverse transition exists and could be shown by a second, backward arrow. The two species of structures are thus equivalent. In practice, one makes no distinction between equivalent species of structures. The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic A-spaces lead to two isomorphic B-spaces. The diagram on Fig.

That is, all directed paths in the diagram with the same start and endpoints lead to the same result. Other diagrams below are also commutative, except for dashed arrows on Fig. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used sometimes, for example, the fine topology ; but these are always specified explicitly, since they are much less notable that the prevalent topology.

A dashed arrow indicates that several transitions are in use and no one is quite prevalent. Two basic spaces are linear spaces also called vector spaces and topological spaces. Linear spaces are of algebraic nature; there are real linear spaces over the field of real numbers , complex linear spaces over the field of complex numbers , and more generally, linear spaces over any field. Every complex linear space is also a real linear space the latter underlies the former , since each real number is also a complex number.

Linear operations, given in a linear space by definition, lead to such notions as straight lines and planes, and other linear subspaces ; parallel lines; ellipses and ellipsoids. However, it is impossible to define orthogonal perpendicular lines, or to single out circles among ellipses, because in a linear space there is no structure like a scalar product that could be used for measuring angles. The dimension of a linear space is defined as the maximal number of linearly independent vectors or, equivalently, as the minimal number of vectors that span the space; it may be finite or infinite.

Two linear spaces over the same field are isomorphic if and only if they are of the same dimension. A n -dimensional complex linear space is also a 2 n -dimensional real linear space. Topological spaces are of analytic nature. Open sets , given in a topological space by definition, lead to such notions as continuous functions , paths, maps; convergent sequences, limits ; interior, boundary, exterior.

However, uniform continuity , bounded sets , Cauchy sequences , differentiable functions paths, maps remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are one-to-one correspondences continuous in both directions. The surface of a cube is homeomorphic to a sphere the surface of a ball but not homeomorphic to a torus. Euclidean spaces of different dimensions are not homeomorphic, which seems evident, but is not easy to prove.

The dimension of a topological space is difficult to define; inductive dimension based on the observation that the dimension of the boundary of a geometric figure is usually one less than the dimension of the figure itself and Lebesgue covering dimension can be used. In the case of a n -dimensional Euclidean space, both topological dimensions are equal to n. Every subset of a topological space is itself a topological space in contrast, only linear subsets of a linear space are linear spaces.

Arbitrary topological spaces, investigated by general topology called also point-set topology are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces "species" of this "type". Every continuous function is bounded on such space. Geometric topology investigates manifolds another "species" of this "type" ; these are topological spaces locally homeomorphic to Euclidean spaces and satisfying a few extra conditions.

Low-dimensional manifolds are completely classified up to homeomorphism. Both the linear and topological structures underlie the linear topological space in other words, topological vector space structure. A linear topological space is both a real or complex linear space and a topological space, such that the linear operations are continuous.

So a linear space that is also topological is not in general a linear topological space. Every finite-dimensional real or complex linear space is a linear topological space in the sense that it carries one and only one topology that makes it a linear topological space. The two structures, "finite-dimensional real or complex linear space" and "finite-dimensional linear topological space", are thus equivalent, that is, mutually underlying.

Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism. The three notions of dimension one algebraic and two topological agree for finite-dimensional real linear spaces.

## Topology, general

In infinite-dimensional spaces, however, different topologies can conform to a given linear structure, and invertible linear transformations are generally not homeomorphisms. It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. Shifting it by a vector external to it, one obtains a n -dimensional affine subspace.

It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All n -dimensional affine spaces are mutually isomorphic. In the words of John Baez , "an affine space is a vector space that's forgotten its origin".

In particular, every linear space is also an affine space. Every point of the affine subspace A is the intersection of A with a one-dimensional linear subspace of L. However, some one-dimensional subspaces of L are parallel to A ; in some sense, they intersect A at infinity. And the affine subspace A is embedded into the projective space as a proper subset.

However, the projective space itself is homogeneous. Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any 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. In a real projective space a straight line is homeomorphic to a circle, therefore compact, in contrast to a straight line in a linear of affine space.

Distances between points are defined in a metric space. Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable , if it underlies a metric space. All manifolds are metrizable. In a metric space, we can define bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge.

Every incomplete space is isometrically embedded, as a dense subset, into a complete space the completion. Every compact metric space is complete; the real line is non-compact but complete; the open interval 0,1 is incomplete. Every Euclidean space is also a complete metric space.

Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points A and C consists of all points B such that the distance between A and C is equal to the sum of two distances, between A and B and between B and C. The Hausdorff dimension related to the number of small balls that cover the given set applies to metric spaces, and can be non-integer especially for fractals. For a n -dimensional Euclidean space, the Hausdorff dimension is equal to n.

Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences or filters or nets , completeness and completion. Every uniform space is also a topological space. Every linear topological space metrizable or not is also a uniform space, and is complete in finite dimension but generally incomplete in infinite dimension. More generally, every commutative topological group is also a uniform space. A non-commutative topological group, however, carries two uniform structures, one left-invariant, the other right-invariant.

A real or complex linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space. Many spaces of sequences or functions are infinite-dimensional Banach spaces. The set of all vectors of norm less than one is called the unit ball of a normed space.

It is a convex, centrally symmetric set, generally not an ellipsoid; for example, it may be a polygon in the plane or, more generally, a polytope in arbitrary finite dimension. The parallelogram law called also parallelogram identity. An inner product space is a real or complex linear space, endowed with a bilinear or respectively sesquilinear form, satisfying some conditions and called an inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid.

Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. Some authors insist that it must be complex, others admit also real Hilbert spaces. Many spaces of sequences or functions are infinite-dimensional Hilbert spaces. Hilbert spaces are very important for quantum theory. All n -dimensional real inner product spaces are mutually isomorphic.

One may say that the n -dimensional Euclidean space is the n -dimensional real inner product space that forgot its origin. Smooth manifolds are not called "spaces", but could be. Every smooth manifold is a topological manifold, and can be embedded into a finite-dimensional linear space. Smooth surfaces in a finite-dimensional linear space are smooth manifolds: for example, the surface of an ellipsoid is a smooth manifold, a polytope is not.

Real or complex finite-dimensional linear, affine and projective spaces are also smooth manifolds. At each one of its points, a smooth path in a smooth manifold has a tangent vector that belongs to the manifold's tangent space at this point. Tangent spaces to an n -dimensional smooth manifold are n -dimensional linear spaces. The differential of a smooth function on a smooth manifold provides a linear functional on the tangent space at each point.

A Riemannian manifold , or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner products satisfying some conditions. Euclidean spaces are also Riemann spaces. Smooth surfaces in Euclidean spaces are Riemann spaces. A hyperbolic non-Euclidean space is also a Riemann space. A curve in a Riemann space has a length, and the length of the shortest curve between two points defines a distance, such that the Riemann space is a metric space.

The angle between two curves intersecting at a point is the angle between their tangent lines. Waiving positivity of inner products on tangent spaces, one obtains pseudo-Riemann spaces , including the Lorentzian spaces that are very important for general relativity. Waiving distances and angles while retaining volumes of geometric bodies one reaches measure theory.

Besides the volume, a measure generalizes the notions of area, length, mass or charge distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory. A "geometric body" of classical mathematics is much more regular than just a set of points. The boundary of the body is of zero volume. Thus, the volume of the body is the volume of its interior, and the interior can be exhausted by an infinite sequence of cubes. In contrast, the boundary of an arbitrary set of points can be of non-zero volume an example: the set of all rational points inside a given cube.

Measure theory succeeded in extending the notion of volume to a vast class of sets, the so-called measurable sets. Indeed, non-measurable sets almost never occur in applications. Measurable sets, given in a measurable space by definition, lead to measurable functions and maps. Baire sets , universally measurable sets , etc, are also used sometimes. Every subset of a measurable space is itself a measurable space. Standard measurable spaces also called standard Borel spaces are especially useful due to some similarity to compact spaces see EoM.

Every bijective measurable mapping between standard measurable spaces is an isomorphism; that is, the inverse mapping is also measurable. And a mapping between such spaces is measurable if and only if its graph is measurable in the product space. Similarly, every bijective continuous mapping between compact metric spaces is a homeomorphism; that is, the inverse mapping is also continuous. And a mapping between such spaces is continuous if and only if its graph is closed in the product space. All uncountable standard measurable spaces are mutually isomorphic.

A measure space is a measurable space endowed with a measure. A Euclidean space with the Lebesgue measure is a measure space. Integration theory defines integrability and integrals of measurable functions on a measure space. Sets of measure 0, called null sets, are negligible.

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A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family finite or not of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinite-dimensional probability measures especially, Gaussian measures , but no infinite-dimensional Lebesgue measures.

Standard probability spaces are especially useful. On a standard probability space a conditional expectation may be treated as the integral over the conditional measure regular conditional probabilities , see also disintegration of measure. Given two standard probability spaces, every homomorphism of their measure algebras is induced by some measure preserving map. Every probability measure on a standard measurable space leads to a standard probability space. The product of a sequence finite or not of standard probability spaces is a standard probability space.

These spaces are less geometric. In particular, the idea of dimension, applicable in one form or another to all other spaces, does not apply to measurable, measure and probability spaces. The theoretical study of calculus, known as mathematical analysis , led in the early 20th century to the consideration of linear spaces of real-valued or complex-valued functions. The earliest examples of these were function spaces , each one adapted to its own class of problems.

These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems. The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space X. This set of functions is a Banach space under pointwise addition and scalar multiplication.

With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on X. An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions of a von Neumann algebra plus a representation and of a measure space are mutually inverse.

Von Neumann then proposed that non-commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray , he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called factors.

Von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any non-negative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors.

A slightly different approach to the geometry of function spaces developed at the same time as von Neumann and Murray's work on the classification of factors. By definition, this is the algebra of continuous complex-valued functions on X that vanish at infinity which loosely means that the farther you go from a chosen point, the closer the function gets to zero with the operations of pointwise addition and multiplication.

Both of these examples are now cases of a field called non-commutative geometry. Non-commutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity.

Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion. This makes it impossible to assign the set of all tilings a topology in the traditional sense.

Another example, and one of great interest within differential geometry , comes from foliations of manifolds. These are ways of splitting the manifold up into smaller-dimensional submanifolds called leaves , each of which is locally parallel to others nearby. The set of all leaves can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be inacessible to the techniques of classical measure theory. However, there is a non-commutative von Neumann algebra associated to the leaf space of a foliation, and once again, this gives an otherwise unintelligible space a good geometric structure.

Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined from the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa. Prior to the s, algebraic geometry worked exclusively over the complex numbers, and the most fundamental variety was projective space. The geometry of projective space is closely related to the theory of perspective , and its algebra is described by homogeneous polynomials.

All other varieties were defined as subsets of projective space. Projective varieties were subsets defined by a set of homogeneous polynomials. At each point of the projective variety, all the polynomials in the set were required to equal zero. The complement of the zero set of a linear polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space. In pursuit of this idea, Weil rewrote the foundations of algebraic geometry, both freeing algebraic geometry from its reliance on complex numbers and introducing abstract algebraic varieties which were not embedded in projective space.

These are now simply called varieties. The type of space that underlies most modern algebraic geometry is even more general than Weil's abstract algebraic varieties.

It was introduced by Alexander Grothendieck and is called a scheme. One of the motivations for scheme theory is that polynomials are unusually structured among functions, and algebraic varieties are consequently rigid. This presents problems when attempting to study degenerate situations. For example, almost any pair of points on a circle determines a unique line called the secant line, and as the two points move around the circle, the secant line varies continuously.

However, when the two points collide, the secant line degenerates to a tangent line. The tangent line is unique, but the geometry of this configurationâ€”a single point on a circleâ€”is not expressive enough to determine a unique line. Studying situations like this requires a theory capable of assigning extra data to degenerate situations.

One of the building blocks of a scheme is a topological space. Topological spaces have continuous functions, but continuous functions are too general to reflect the underlying algebraic structure of interest. The other ingredient in a scheme, therefore, is a sheaf on the topological space, called the "structure sheaf".

On each open subset of the topological space, the sheaf specifies a collection of functions, called "regular functions". The topological space and the structure sheaf together are required to satisfy conditions that mean the functions come from algebraic operations. Like manifolds, schemes are defined as spaces that are locally modeled on a familiar space.

In the case of manifolds, the familiar space is Euclidean space. For a scheme, the local models are called affine schemes. Affine schemes provide a direct link between algebraic geometry and commutative algebra.