Information about Scalar Product

For the scalar product or dot product of spatial vectors, see dot product.

In mathematics, an inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry. The additional structure associates to each pair of vectors in the space a number which is called the inner product (also called a scalar product) of the vectors. 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. It also allows introduction of the concept of orthogonality between vectors. Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.

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.

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

Definition

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.

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

satisfying the following axioms for all :

• Conjugate symmetry:

:

This condition implies that , because .

(Conjugation is also often written with an asterisk, as in , as is the conjugate transpose.)

• Linearity in the first variable:

:

:

By combining these with conjugate symmetry, we get:

:

:

So is actually a sesquilinear form.

• Nonnegativity:

:

(This makes sense because for all .)

• Nondegeneracy:

: implies .

Hence, the inner product is a nonnegative, nondegenerate Hermitian form.

The property of an inner product space that

: and 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 bilinearity.

Remark. Most mathematical authors require an inner product to be linear in the first argument and conjugate-linear 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 bra-ket 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.

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.

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

More generally any Euclidean space Rⁿ with the dot product is an inner product space

The general form of an inner product on Cⁿ is given by:

with M any symmetric 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.

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

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

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.

• Orthogonality: 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 [0, +π]. This is in analogy to the 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.

• Triangle inequality: for x, y elements of V

:

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.

• Parallelogram law:

:

• Pythagorean theorem: Whenever x, y are in V and <x, y> = 0, then

:

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, ..., x_n are orthogonal vectors, that is, <x_j, x_k> = 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 {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.

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 of finite dimension V is an orthonormal sequence whose algebraic span is V. This definition of orthonormal basis does not generalise conveniently to the case of infinite dimensions, where the concept (properly formulated) is of major importance. 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.

The Gram-Schmidt 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

By the Gram-Schmidt orthonormalization process, one shows:

Theorem. Any separable inner product space V has an orthonormal basis.

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 → l² 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:

Theorem. Let V be the inner product space . Then the sequence (indexed on set of all integers) of continuous functions

is an orthonormal basis of the space with the L² 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 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:

• 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.

  Degenerate inner products

  If V is a vector space and < , > a semi-definite 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 semi-norm.) We can produce an inner product space by considering the quotient W = V/{ x : ||x|| = 0}. The sesquilinear form < , > factors through W.
  
  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.

  See also

  • Outer product
  • Exterior algebra
  • bilinear form
  • dual space
  • dual pair
  • biorthogonal system
  • Fubini-Study metric
  • Energetic space

  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
  dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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  Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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  In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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  Structure is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic
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  Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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  dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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  angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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  Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth
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  Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.
  
  An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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  dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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  Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier
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  Hilbert space, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.
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  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.
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  scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.
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  In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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  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 = −1.
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  In mathematics, a definite bilinear form is a bilinear form B such that
  
  

  B(x, x)

  
  
  has a fixed sign (positive or negative) when x is not 0.
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  In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that the map from V to V* (the dual space of V) given by v f(-,v) is not a bijection.
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  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.
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  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.
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  A symmetric bilinear form is, as the name implies, a bilinear form on a vector space that is symmetric. They are of great importance in the study of orthogonal polarities and quadrics.
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  In mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F, where F is the field of scalars.
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  axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.
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  complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number

  (where and are real numbers) is
  
  

  
  
  The complex conjugate is also very commonly denoted by .
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  conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A^* obtained from A
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  prevew not available
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  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.
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  In mathematics, a bilinear map is a function which is linear in both of its arguments. An example of such a map is multiplication of integers.

  Definition

  Let V, W and X be three vector spaces over the same base field F.
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  Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics.
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  quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
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