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Encyclopedia > Inner product

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, a famous Greek mathematician known as the father of geometry, is shown here in detail from 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. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... 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 pre-Hilbert 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 3-dimensional vectors with real-valued 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 generalization of Euclidean space which is not restricted to finite dimensions. ...


Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used.

Contents


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. This article presents the essential definitions. ... 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 are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...


Formally, an inner product space is a vector space V over the field F together with a positive-definite nondegenerate sesquilinear form, called an inner product. For real vector spaces, this is actually a positive-definite 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 non-zero 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 × V → F which is linear in both arguments. ...

satisfying the following axioms: An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ...

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:
  • Nonnegativity:
(This makes sense because for all .)
  • Nondegeneracy:
The map from V to the dual space V* given by is an isomorphism. For a finite-dimensional 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 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 m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... 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 mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×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. Many mathematical authors require an inner product to be linear in the first argument and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the bra-ket notation used by physicists in quantum mechanics and is now often 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. Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ...


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 non-negativity 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 b It follows from these axioms... 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 non-negative semi-definite sesquilinear forms. This means that <x, x> is only required to be non-negative. 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 Rn 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 Cn is given by:

with M any positive-definite 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 weighted-sum version of the dot product, with positive weights. In linear algebra, a positive-definite 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 m-by-n matrix A with complex entries is the n-by-m 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 generalization of Euclidean space which is not restricted to finite dimensions. ... 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 { fk }k where

  • fk(t) is 1 for t in the subinterval [0, 1/2]
  • fk(t) is 0 for t in the subinterval [1/2 + 1/k, 1]
  • fk 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 3-dimensional vectors with real-valued 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:

  • Cauchy-Schwarz 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 Cauchy-Bunyakowski-Schwarz 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
Enlarge
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 Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero 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 two-dimensional Euclidean space. Correspondingly, we will say that non-zero 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 x1, ..., xn are orthogonal vectors, that is, <xj, xk> = 0 for distinct indices j, k, then
In view of the Cauchy-Schwarz 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 {xk} 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 Cauchy-Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy-Bunyakovski-Schwarz 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. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Homogeneity in physics In physics, homogeneity is the quality of having all properties independent of the position. ... 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 3-dimensional vectors with real-valued 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 generalization of Euclidean space which is not restricted to finite dimensions. ... 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 Pythagorass theorem is a relation in Euclidean geometry between 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 {ek}k is orthonormal if and only if it is orthogonal and each ek 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 Gram-Schmidt process is a canonical procedure that takes a linearly independent sequence {vk}k on an inner product space and produces an orthonormal sequence {ek}k such that for each n In mathematics and numerical analysis, the Gram-Schmidt 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 Gram-Schmidt 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 {ek}k an orthonormal basis of V. Then the map

is an isometric linear map Vl2 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 l2 is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series: Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... 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 generalization of Euclidean space which is not restricted to finite dimensions. ...


Theorem. Let V be the inner product space C[ − π,π]. Then the sequence (indexed on set of all integers) of continuous functions

ek(t) = (2π) − 1 / 2eikt

is an orthonormal basis of the space C[ − π,π] with the L2 inner product. The mapping

is an isometric linear map with dense image.


Orthogonality of the sequence {ek}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 non-negative 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 one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a morphism is an abstraction of a structure-preserving 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 one-to-one correspondence is a function that is both injective (one-to-one) 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 a collection 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 semi-definite sesquilinear form, then the function ||x|| = <xx>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 semi-norm.) 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 Gelfand-Naimark-Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite 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 × V → 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 (1×n) and column vectors (n×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. ...

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

  Results from FactBites:
 
Reference.com/Encyclopedia/Inner product space (1907 words)
Inner products allow the rigorous introduction of intuitive geometrical notions such as the angle between vectors or length of vectors in spaces of all dimensions.
Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.
Using the norm associated to the inner product, one has the notion of dense subset, and the appropriate definition of orthonormal basis is that the algebraic span (subspace of finite linear combinations of basis vectors) should be dense.
Inner product space - Wikipedia, the free encyclopedia (1808 words)
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.
Hence, the inner product is a Hermitian form.
  More results at FactBites »

 
 

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