In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
This article is about angles in geometry. ...
Length is the long dimension of any object. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
An inner product space is sometimes also called a preHilbert space, since its completion with respect to the metric induced by its inner product is a Hilbert space. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
Link titleIn mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used. Definitions
In the following article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. See below. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ...
In mathematics, the real numbers may be described informally in several different ways. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
Formally, an inner product space is a vector space V over the field F together with a positivedefinite nondegenerate sesquilinear form, called an inner product. For real vector spaces, this is actually a positivedefinite nondegenerate symmetric bilinear form. Thus the inner product is a map In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some nonzero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...
In multilinear algebra, a multilinear form is a map of the type , where V is a vector space over the field K, that is separately linear in each its N variables. ...
In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...
In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ...
satisfying the following axioms: This article does not cite its references or sources. ...

 This condition implies that for all , because .
 (Conjugation is also often written with an asterisk, as in , as is the conjugate transpose.)

 By combining these with conjugate symmetry, we get:
So is actually a Sesquilinear form. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
In mathematics, the conjugate transpose or adjoint of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...
The word linear comes from the Latin word linearis, which means created by lines. ...
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...

 (This makes sense because for all .)
 The map from V to the dual space V* given by is an isomorphism. For a finitedimensional vector space, it suffices to check injectivity:
 Hence, the inner product is a Hermitian form.
The property of an inner product space V that In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
An injective function. ...
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...

 and
for all is known as additivity. Note that if F=R, then the conjugate symmetry property is simply symmetry of the inner product, i.e. 
In this case, sesquilinearity becomes standard linearity. The word linear comes from the Latin word linearis, which means created by lines. ...
Remark. Most mathematical authors require an inner product to be linear in the first argument and conjugatelinear in the second argument, in agreement with the convention adopted above. Many physicists adopt the opposite convention. This change is immaterial, but the opposite definition provides a smoother connection to the braket notation used by physicists in quantum mechanics (in that it allows scalars to come directly out of kets, which represent vectors, while making scalars become conjugated when extracted from bras, which represent linear functionals) and is now occasionally used by mathematicians as well. Some authors adopt the convention that < , > is linear in the first component while <  > is linear in the second component, although this is by no means universal. For instance (Emch [1972]) does not follow this convention. Braket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...
Fig. ...
There are various technical reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield (in order for nonnegativity to make sense) and therefore has to have characteristic equal to 0. This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. It has been suggested that this article or section be merged into scalar. ...
In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
In some cases we need to consider nonnegative semidefinite sesquilinear forms. This means that <x, x> is only required to be nonnegative. We show how to treat these below.
Examples A trivial example are the real numbers with the standard multiplication as the inner product Please refer to Real vs. ...
More generally any Euclidean space R^{n} with the dot product is an inner product space In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
The general form of an inner product on C^{n} is given by: with M any positivedefinite matrix, and x^{*} the conjugate transpose of x. For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Apart from an orthogonal transformation it is a weightedsum version of the dot product, with positive weights. In linear algebra, a positivedefinite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...
In mathematics, the conjugate transpose or adjoint of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ...
A scale factor is a number which scales some quantity. ...
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ...
The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C[a, b] of continuous complex valued functions on the interval [a,b]. The inner product is In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...
This space is not complete; consider for example, for the interval [0,1] the sequence of functions { f_{k} }_{k} where  f_{k}(t) is 1 for t in the subinterval [0, 1/2]
 f_{k}(t) is 0 for t in the subinterval [1/2 + 1/k, 1]
 f_{k} is affine in [1/2, 1/2 + 1/k]
This sequence is a Cauchy sequence which does not converge to a continuous function.
Norms on inner product spaces Inner product spaces have a naturally defined norm Link titleIn mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:  CauchySchwarz inequality: for x, y elements of V

 with equality if and only if x and y are linearly dependent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the CauchyBunyakowskiSchwarz inequality.
 Because of its importance, its short proof should be noted. To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Thus we may let

 and it follows that

 multiplying out, the result follows.
Geometric interpretation of inner product  The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the CauchySchwarz inequality is that it justifies defining the angle between two nonzero vectors x and y (at least in the case F = R) by the identity
 We assume the value of the angle is chosen to be in the interval (−π, +π]. This is in analogy to the familiar situation in twodimensional Euclidean space. Correspondingly, we will say that nonzero vectors x, y of V are orthogonal if and only if their inner product is zero.
 Homogeneity: for x an element of V and r a scalar

 The homogeneity property is completely trivial to prove.

 The last two properties show the function defined is indeed a norm.
 Because of the triangle inequality and because of axiom 2, we see that · is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. Every inner product V space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by V and is constructed by completing V.


 The proofs of both of these identities require only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component. The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated is deeper than the version given above.
 An easy induction on the Pythagorean theorem yields:
 If x_{1}, ..., x_{n} are orthogonal vectors, that is, <x_{j}, x_{k}> = 0 for distinct indices j, k, then

 In view of the CauchySchwarz inequality, we also note that <·,·> is continuous from V × V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:
 Parseval's identity: Suppose V is a complete inner product space. If {x_{k}} are mutually orthogonal vectors in V then

 provided the infinite series on the left is convergent. Completeness of the space is needed to ensure that the sequence of partial sums

 which is easily shown to be a Cauchy sequence is convergent.
In mathematics, the CauchySchwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the CauchyBunyakovskiSchwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in...
In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...
Bigger picture illustrating inner product File links The following pages link to this file: Inner product space Categories: GFDL images ...
Bigger picture illustrating inner product File links The following pages link to this file: Inner product space Categories: GFDL images ...
This article is about angles in geometry. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...
In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...
Link titleIn mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
In mathematics, the term dense has at least three different meanings. ...
// The parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. ...
In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...
Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ...
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
For a discussion of convergence and convergent series, see limit (mathematics). ...
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
Orthonormal sequences A sequence {e_{k}}_{k} is orthonormal if and only if it is orthogonal and each e_{k} has norm 1. An orthonormal basis for an inner product space V is an orthonormal sequence whose algebraic span is V. It has been suggested that this article or section be merged with Logical biconditional. ...
The GramSchmidt process is a canonical procedure that takes a linearly independent sequence {v_{k}}_{k} on an inner product space and produces an orthonormal sequence {e_{k}}_{k} such that for each n In mathematics and numerical analysis, the GramSchmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
By the GramSchmidt orthonormalization process, one shows: Theorem. Any separable inner product space V has an orthonormal basis. In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...
Parseval's identity leads immediately to the following theorem: Theorem. Let V be a separable inner product space and {e_{k}}_{k} an orthonormal basis of V. Then the map is an isometric linear map V → l^{2} with a dense image. This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided l^{2} is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series: The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
In the mathematical subfield of numerical analysis, a trigonometric polynomial is a finite linear linear combination of sin(nx) and cos(nx) with n a natural number. ...
In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
Theorem. Let V be the inner product space C[ − π,π]. Then the sequence (indexed on set of all integers) of continuous functions  e_{k}(t) = (2π) ^{− 1 / 2}e^{ikt}
is an orthonormal basis of the space C[ − π,π] with the L^{2} inner product. The mapping is an isometric linear map with dense image. Orthogonality of the sequence {e_{k}}_{k} follows immediately from the fact that if k ≠ j, then Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on [ − π,π] with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.
Operators on inner product spaces Several types of linear maps A from an inner product space V to an inner product space W are of relevance: The word linear comes from the Latin word linearis, which means created by lines. ...
 Continuous linear maps, i.e. A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of nonnegative reals {Ax}, where x ranges over the closed unit ball of V, is bounded.
 Symmetric linear operators, i.e. A is linear and <Ax, y> = <x, A y> for all x, y in V.
 Isometries, i.e. A is linear and <Ax, Ay> = <x, y> for all x, y in V, or equivalently, A is linear and Ax = x for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
 Isometrical isomorphisms, i.e. A is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).
From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces. In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ...
In mathematics, an injective function (or onetoone function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a morphism is an abstraction of a structurepreserving process between two mathematical structures. ...
In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In mathematics, a bijection, bijective function, or onetoone correspondence is a function that is both injective (onetoone) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...
Normal may refer to: Normal (behavior) Normal (mathematics), a group of mathematical concepts Surface normal, a line or vector perpendicular to a surface Normal (movie), a 2003 film directed by Jane Anderson Normal, Alabama, home to Alabama Agricultural and Mecahnical University Normal, Illinois, a town in the United States Normal...
Degenerate inner products If V is a vector space and < , > a semidefinite sesquilinear form, then the function x = <x, x>^{1/2} makes sense and satisfies all the properties of norm except that x = 0 does not imply x = 0. (Such a functional is then called a seminorm.) We can produce an inner product space by considering the quotient W = V/{ x : x = 0}. The sesquilinear form < , > factors through W. In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...
This construction is used in numerous contexts. The GelfandNaimarkSegal construction is a particularly important example of the use of this technique. Another example is the representation of semidefinite kernels on arbitrary sets. In functional analysis, given a C*algebra A, the GNS construction establishes a correspondence between cyclic *representations of A and certain linear functionals on A (called states). ...
See also Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ...
In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ...
In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ...
In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form. ...
In mathematics, a biorthogonal system in a pair of topological vector spaces E and F that are in duality is a pair of indexed subsets vi in E and wi in F such that <vi,wj> = δij with the Kronecker delta. ...
In mathematics, a KÃ¤hler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...
References  S. Axler, Linear Algebra Done Right, Springer, 2004
 G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley Interscience, 1972.
 N. Young, An Introduction to Hilbert Spaces, Cambridge University Press, 1988
