Euclidean space

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Euclidean space is the fundamental space of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, but in modern
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
there are Euclidean spaces of any positive integer
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...
mathematician
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in his ''Elements'', with the great innovation of '' proving'' all properties of the space as
theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the th ...
s, by starting from a few fundamental properties, called '' postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...
). After the introduction at the end of 19th century of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real -space $\R^n,$ equipped with the dot product. An isomorphism from a Euclidean space to $\R^n$ associates with each point an -tuple of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s which locate that point in the Euclidean space and are called the '' Cartesian coordinates'' of that point.

Definition

History of the definition

Euclidean space was introduced by
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
as an abstraction of our physical space. Their great innovation, appearing in Euclid's ''Elements'' was to build and '' prove'' all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry. In 1637,
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...
introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
was a major change in point of view, as, until then, the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension , using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension. Despite the wide use of Descartes' approach, which was called
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.

Motivation of the modern definition

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is
rotation Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A t ...
around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s) of the plane should be considered equivalent ( congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below). In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in theories, Euclidean space is an
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. T ...

detached from actual physical locations, specific
reference frames In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...

, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of
units of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric units, used in every country globally. In the United States the U.S. customary uni ...
and other physical dimensions: the distance in a "mathematical" space is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

, not something expressed in inches or metres. The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of The gospel, ...
, the ''space of translations'' which is equipped with an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. The action of translations makes the space an
affine space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles. The set $\R^n$ of -tuples of real numbers equipped with the dot product is a Euclidean space of dimension . Conversely, the choice of a point called the ''origin'' and an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit ...
of the space of translations is equivalent with defining an
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...

between a Euclidean space of dimension and $\R^n$ viewed as a Euclidean space. It follows that everything that can be said about a Euclidean space can also be said about $\R^n.$ Therefore, many authors, especially at elementary level, call $\R^n$ the ''standard Euclidean space'' of dimension , or simply ''the'' Euclidean space of dimension . A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of $\R^n$ is that it is often preferable to work in a ''coordinate-free'' and ''origin-free'' manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no origin nor any basis in the physical world.

Technical definition

A is a finite-dimensional
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
over the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. A Euclidean space is an
affine space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' for distinguishing them from Euclidean vector spaces. If is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted $\overrightarrow E.$ The ''dimension'' of a Euclidean space is the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of its associated vector space. The elements of are called ''points'' and are commonly denoted by capital letters. The elements of $\overrightarrow E$ are called ''
Euclidean vector In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s'' or ''
free vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has Magnitude (mathematics), magnitude (or euclidean norm, length) and Direction ( ...
s''. They are also called ''translations'', although, properly speaking, a
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
is the
geometric transformation In mathematics, a geometric transformation is any bijection of a Set (mathematics), set to itself (or to another such set) with some salient geometry, geometrical underpinning. More specifically, it is a function (mathematics), function whose Do ...
resulting of the
action Action may refer to: * Action (narrative) In literature, action is the physical movement of the Character (arts), characters. Action as a literary mode "Action is the Mode (literature), mode hat A hat is a head covering which is worn for ...
of a Euclidean vector on the Euclidean space. The action of a translation on a point provides a point that is denoted . This action satisfies $P+(v+w)= (P+v)+w.$ Note: The second in the left-hand side is a vector addition; all other denote an action of a vector on a point. This notation is not ambiguous, as, for distinguishing between the two meanings of , it suffices to look on the nature of its left argument. The fact that the action is free and transitive means that for every pair of points there is exactly one
displacement vector In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
such that . This vector is denoted or $\overrightarrow .$ As previously explained, some of the basic properties of Euclidean spaces result of the structure of affine space. They are described in and its subsections. The properties resulting from the inner product are explained in and its subsections.

Prototypical examples

For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space. A typical case of Euclidean vector space is $\R^n$ viewed as a vector space equipped with the dot product as an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space of dimension , the choice of a point, called an ''origin'' and an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit ...
of $\overrightarrow E$ defines an isomorphism of Euclidean spaces from to $\R^n.$ As every Euclidean space of dimension is isomorphic to it, the Euclidean space $\R^n$ is sometimes called the ''standard Euclidean space'' of dimension .

Affine structure

Some basic properties of Euclidean spaces depend only of the fact that a Euclidean space is an
affine space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.

Subspaces

Let be a Euclidean space and $\overrightarrow E$ its associated vector space. A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of is a subset of such that $\overrightarrow F = \left\$ as the associated vector space of is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flat (geometry), flats and affine subspaces. In the case of vector spaces o ...
(vector subspace) of $\overrightarrow E.$ A Euclidean subspace is a Euclidean space with $\overrightarrow F$ as the associated vector space. This linear subspace $\overrightarrow F$ is also called the ''direction'' of . If is a point of then $F = \left\.$ Conversely, if is a point of and $\overrightarrow V$ is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flat (geometry), flats and affine subspaces. In the case of vector spaces o ...
of $\overrightarrow E,$ then $P + V = \left\$ is a Euclidean subspace of direction $\overrightarrow V$. (The associated vector space of this subspace is $\overrightarrow V$.) A Euclidean vector space $\overrightarrow E$ (that is, a Euclidean space that is equal to $\overrightarrow E$) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

Lines and segments

In a Euclidean space, a ''line'' is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form $\left\,$ where and are two distinct points of the Euclidean space as a part of the line. It follows that ''there is exactly one line that passes through (contains) two distinct points.'' This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through and is $\left\,$ where is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for ; this allows simplifying the preceding formula into $\left\.$ A standard convention allows using this formula in every Euclidean space, see . The ''
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...

'', or simply ''segment'', joining the points and is the subset of points such that in the preceding formulas. It is denoted or ; that is $PQ = QP = \left\.$

Parallelism

Two subspaces and of the same dimension in a Euclidean space are ''parallel'' if they have the same direction (i.e., the same associated vector space). Equivalently, they are parallel, if there is a translation vector that maps one to the other: $T= S+v.$ Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is $P + \overrightarrow S.$ In the case where is a line (subspace of dimension one), this property is Playfair's axiom. It follows that in a Euclidean plane, two lines either meet in one point or are parallel. The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

Metric structure

The vector space $\overrightarrow E$ associated to a Euclidean space is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. This implies a
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field (mathematics), field of Scalar (mathematics), scalars such that the order of the two vectors does not affect the value of ...
$\begin \overrightarrow E \times \overrightarrow E &\to \R\\ (x,y)&\mapsto \langle x,y \rangle \end$ that is positive definite (that is $\langle x,x \rangle$ is always positive for ). The inner product of a Euclidean space is often called ''dot product'' and denoted . This is specially the case when a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
has been chosen, as, in this case, the inner product of two vectors is the dot product of their
coordinate vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...
s. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is $\langle x,y \rangle$ will be denoted in the remainder of this article. The Euclidean norm of a vector is $\, x\, = \sqrt .$ The inner product and the norm allows expressing and proving
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
and
topological In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

properties of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
. The next subsection describe the most fundamental ones. ''In these subsections,'' ''denotes an arbitrary Euclidean space, and $\overrightarrow E$ denotes its vector space of translations.''

Distance and length

The ''distance'' (more precisely the ''Euclidean distance'') between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is $d(P,Q) = \Bigl\, \overrightarrow \vphantom\Bigr\, .$ The ''length'' of a segment is the distance between its endpoints ''P'' and ''Q''. It is often denoted $, PQ,$. The distance is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
, as it is positive definite, symmetric, and satisfies the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...

$d(P,Q)\le d(P,R) + d(R, Q).$ Moreover, the equality is true if and only if a point belongs to the segment . This inequality means that the length of any edge of a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

is smaller than the sum of the lengths of the other edges. This is the origin of the term ''triangle inequality''. With the Euclidean distance, every Euclidean space is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
.

Orthogonality

Two nonzero vectors and of $\overrightarrow E$ (the associated vector space of a Euclidean space ) are ''perpendicular'' or ''orthogonal'' if their inner product is zero: $u \cdot v =0$ Two linear subspaces of $\overrightarrow E$ are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector. Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said ''perpendicular''. Two segments and that share a common endpoint are ''perpendicular'' or ''form a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

'' if the vectors $\overrightarrow$ and $\overrightarrow$ are orthogonal. If and form a right angle, one has $, BC, ^2 = , AB, ^2 + , AC, ^2.$ This is the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product: $\begin , BC, ^2 &= \overrightarrow \cdot \overrightarrow \\ &=\left(\overrightarrow +\overrightarrow \right ) \cdot \left(\overrightarrow +\overrightarrow \right)\\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot \overrightarrow -2 \overrightarrow \cdot \overrightarrow \\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot\overrightarrow \\ &=, AB, ^2 + , AC, ^2. \end$ Here, $\overrightarrow \cdot \overrightarrow = 0$ is used since these two vectors are orthogonal.

Angle

The (non-oriented) ''angle'' between two nonzero vectors and in $\overrightarrow E$ is $\theta = \arccos\left(\frac\right)$ where is the
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch (mathematics), branch of that Function (mathematics), function, so that it is Single-valued function, single-val ...
of the function. By
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used Inequality (mathematics), inequalities in mathematics. The inequality for sums was published by . The c ...
, the argument of the arccosine is in the interval . Therefore is real, and (or if angles are measured in degrees). Angles are not useful in a Euclidean line, as they can be only 0 or . In an
oriented In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
Euclidean plane, one can define the ''oriented angle'' of two vectors. The oriented angle of two vectors and is then the opposite of the oriented angle of and . In this case, the angle of two vectors can have any value modulo an integer multiple of . In particular, a equals the negative angle . The angle of two vectors does not change if they are by positive numbers. More precisely, if and are two vectors, and and are real numbers, then $\operatorname(\lambda x, \mu y)= \begin \operatorname(x, y) \qquad\qquad \text \lambda \text \mu \text\\ \pi - \operatorname(x, y)\qquad \text. \end$ If , , and are three points in a Euclidean space, the angle of the segments and is the angle of the vectors $\overrightarrow$ and $\overrightarrow .$ As the multiplication of vectors by positive numbers do not change the angle, the angle of two
half-line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
s with initial point can be defined: it is the angle of the segments and , where and are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point. The angle of two lines is defined as follows. If is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either or . One of these angles is in the interval , and the other being in . The ''non-oriented angle'' of the two lines is the one in the interval . In an oriented Euclidean plane, the ''oriented angle'' of two lines belongs to the interval .

Cartesian coordinates

Every Euclidean vector space has an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit ...
(in fact, infinitely many in dimension higher than one, and two in dimension one), that is a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
$\left(e_1, \dots, e_n\right)$ of
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s ($\, e_i\, = 1$) that are pairwise orthogonal ($e_i\cdot e_j = 0$ for ). More precisely, given any
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
$\left(b_1, \dots, b_n\right),$ the
Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for Orthonormal basis, orthonormalizing a set of vector (geometry), vectors in an inner product space, most commonly the Euclidean space eq ...

computes an orthonormal basis such that, for every , the
linear span In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s of $\left(e_1, \dots, e_i\right)$ and $\left(b_1, \dots, b_i\right)$ are equal. Given a Euclidean space , a ''Cartesian frame'' is a set of data consisting of an orthonormal basis of $\overrightarrow E,$ and a point of , called the ''origin'' and often denoted . A Cartesian frame $\left(O, e_1, \dots, e_n\right)$ allows defining Cartesian coordinates for both and $\overrightarrow E$ in the following way. The Cartesian coordinates of a vector of $\overrightarrow E$ are the coefficients of on the orthonormal basis $e_1, \dots, e_n.$ For example, the Cartesian coordinates of a vector $v$ on an orthonormal basis $\left(e_1,e_2,e_3\right)$ (that may be named as $\left(x,y,z\right)$ as a convention) in a 3-dimensional Euclidean space is $\left(\alpha_1,\alpha_2,\alpha_3\right)$ if $v = \alpha_1 e_1 + \alpha_2 e_2 + \alpha_3 e_3$. As the basis is orthonormal, the -th coefficient $\alpha_i$ is equal to the dot product $v\cdot e_i.$ The Cartesian coordinates of a point of are the Cartesian coordinates of the vector $\overrightarrow .$

Other coordinates

As a Euclidean space is an
affine space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define affine coordinates, sometimes called ''skew coordinates'' for emphasizing that the basis vectors are not pairwise orthogonal. An affine basis of a Euclidean space of dimension is a set of points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point. Many other coordinates systems can be defined on a Euclidean space of dimension , in the following way. Let be a
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
(or, more often, a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
) from a dense
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
of to an open subset of $\R^n.$ The ''coordinates'' of a point of are the components of . The
polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point (mathematics), point on a plane (mathematics), plane is determined by a distance from a reference point and an angle from a reference direction ...

(dimension 2) and the
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

and
cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder ...

coordinate systems (dimension 3) are defined this way. For points that are outside the domain of , coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the
antimeridian The 180th meridian or antimeridian is the meridian 180° both east and west of the prime meridian A prime meridian is an arbitrary meridian (geography), meridian (a line of longitude) in a geographic coordinate system at which longitude ...
, the longitude passes discontinuously from –180° to +180°. This way of defining coordinates extends easily to other mathematical structures, and in particular to
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

s.

Isometries

An
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. The word isometry is derived from the Ancient Greek: ἴσος ' ...
between two
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s is a bijection preserving the distance, that is $d(f(x), f(y))= d(x,y).$ In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm $\, f(x)\, = \, x\, ,$ since the norm of a vector is its distance from the zero vector. It preserves also the inner product $f(x)\cdot f(y)=x\cdot y,$ since $x \cdot y=\frac 1 2 \left(\, x+y\, ^2-\, x\, ^2-\, y\, ^2\right).$ An isometry of Euclidean vector spaces is a
linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
. An isometry $f\colon E\to F$ of Euclidean spaces defines an isometry $\overrightarrow f \colon \overrightarrow E \to \overrightarrow F$ of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if and are Euclidean spaces, , , and $\overrightarrow f\colon \overrightarrow E\to \overrightarrow F$ is an isometry, then the map $f\colon E\to F$ defined by $f(P)=O' + \overrightarrow f\left(\overrightarrow\right)$ is an isometry of Euclidean spaces. It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.

Isometry with prototypical examples

If is a Euclidean space, its associated vector space $\overrightarrow E$ can be considered as a Euclidean space. Every point defines an isometry of Euclidean spaces $P\mapsto \overrightarrow ,$ which maps to the zero vector and has the identity as associated linear map. The inverse isometry is the map $v\mapsto O+v.$ A Euclidean frame allows defining the map $\begin E&\to \R^n\\ P&\mapsto \left(e_1\cdot \overrightarrow , \dots, e_n\cdot\overrightarrow \right), \end$ which is an isometry of Euclidean spaces. The inverse isometry is $\begin \R^n&\to E \\ (x_1\dots, x_n)&\mapsto \left(O+x_1e_1+ \dots + x_ne_n\right). \end$ ''This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.'' This justifies that many authors talk of $\R^n$ as ''the'' Euclidean space of dimension .

Euclidean group

An isometry from a Euclidean space onto itself is called ''Euclidean isometry'', ''Euclidean transformation'' or ''rigid transformation''. The rigid transformations of a Euclidean space form a group (under composition), called the ''Euclidean group'' and often denoted of . The simplest Euclidean transformations are translation (mathematics), translations $P \to P+v.$ They are in bijective correspondence with vectors. This is a reason for calling ''space of translations'' the vector space associated to a Euclidean space. The translations form a normal subgroup of the Euclidean group. A Euclidean isometry of a Euclidean space defines a linear isometry $\overrightarrow f$ of the associated vector space (by ''linear isometry'', it is meant an isometry that is also a linear map) in the following way: denoting by the vector $\overrightarrow$, if is an arbitrary point of , one has $\overrightarrow f(\overrightarrow )= f(P)-f(O).$ It is straightforward to prove that this is a linear map that does not depend from the choice of The map $f \to \overrightarrow f$ is a group homomorphism from the Euclidean group onto the group of linear isometries, called the orthogonal group. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group. The isometries that fix a given point form the stabilizer subgroup of the Euclidean group with respect to . The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group. Let be a point, an isometry, and the translation that maps to . The isometry $g=t^\circ f$ fixes . So $f= t\circ g,$ and ''the Euclidean group is the semidirect product of the translation group and the orthogonal group.'' The special orthogonal group is the normal subgroup of the orthogonal group that preserves orientation (vector space), handedness. It is a subgroup of index (group theory), index two of the orthogonal group. Its inverse image by the group homomorphism $f \to \overrightarrow f$ is a normal subgroup of index two of the Euclidean group, which is called the ''special Euclidean group'' or the ''displacement group''. Its elements are called ''rigid motions'' or ''displacements''. Rigid motions include the identity function, identity, translations, rotations (the rigid motions that fix at least a point), and also screw axis, screw motions. Typical examples of rigid transformations that are not rigid motions are reflections, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame. As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection , every rigid transformation that is not a rigid motion is the product of and a rigid motion. A glide reflection is an example of a rigid transformation that is not a rigid motion or a reflection. All groups that have been considered in this section are Lie groups and algebraic groups.

Topology

The Euclidean distance makes a Euclidean space a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, and thus a topological space. This topology is called the Euclidean topology. In the case of $\mathbb R^n,$ this topology is also the product topology. The open sets are the subsets that contains an open ball around each of their points. In other words, open balls form a base (topology), base of the topology. The topological dimension of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not homeomorphic. Moreover, the theorem of invariance of domain asserts that a subset of a Euclidean space is open (for the subspace topology) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension. Euclidean spaces are complete metric, complete and locally compact. That is, a closed subset of a Euclidean space is compact if it is bounded set, bounded (that is, contained in a ball). In particular, closed balls are compact.

Axiomatic definitions

The definition of Euclidean spaces that has been described in this article differs fundamentally of
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
. Two different approaches have been used. Felix Klein suggested to define geometries through their symmetries. The presentation of Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries. On the other hand, David Hilbert proposed a set of Hilbert's axioms, axioms, inspired by Euclid's postulates. They belong to synthetic geometry, as they do not involve any definition of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s (see Birkhoff's axioms and Tarski's axioms). In ''Geometric Algebra (book), Geometric Algebra'', Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence (geometry), congruence is an equivalence relation on segments. One can thus define the ''length'' of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert.

Usage

Since
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
, Euclidean space is used for modeling shapes in the physical world. It is thus used in many sciences such as
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, or technical drawing. Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration space (physics), configuration spaces of physical systems. Beside
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

is a space that is locally approximated by Euclidean spaces. Most
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
can be modeled by a manifold, and embedding, embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematical objects that are ''a priori'' not of a geometrical nature. An example among many is the usual representation of Graph (discrete mathematics), graphs.

Other geometric spaces

Since the introduction, at the end of 19th century, of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
is consistent (which cannot be proved).

Affine space

A Euclidean space is an affine space equipped with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field (mathematics), field, they allow doing geometry in other contexts. As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the complex numbers as an extension of Euclidean spaces. For example, a circle and a line (geometry), line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic variety, affine algebraic varieties. Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals." Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography.

Projective space

Originally, projective spaces have been introduced by adding "points at infinity" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two coplanar lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the vector lines in a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of dimension one more. As for affine spaces, projective spaces are defined over any field (mathematics), field, and are fundamental spaces of algebraic geometry.

Non-Euclidean geometries

''Non-Euclidean geometry'' refers usually to geometrical spaces where the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...
is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistency, consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theory, axiomatic theories in mathematics.

Curved spaces

A
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately. Distances and angles can be defined on a smooth manifold by providing a smooth function, smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent. Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, geodesics are great circle, arcs of great circle, which are called orthodromes in the context of navigation. More generally, the spaces of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
can be realized as Riemannian manifolds.

Pseudo-Euclidean space

An
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of a real vector space is a positive definite bilinear form, and so characterized by a Bilinear form#Derived quadratic form, positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite quadratic form, indefinite). A fundamental example of such a space is the Minkowski space, which is the space-time of Albert Einstein, Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form $x^2+y^2+z^2-t^2,$ where the last coordinate (''t'') is temporal, and the other three (''x'', ''y'', ''z'') are spatial. To take gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The Curvature of Riemannian manifolds, curvature of this manifold at a point is a function of the value of the gravitational field at this point.