Inner product space

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Geometric interpretation of the angle between two vectors defined using an inner product Inner-product-angle.png
Geometric interpretation of the angle between two vectors defined using an inner product
Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite. Product Spaces Drawing (1).png
Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.

In mathematics, an inner product space or a Hausdorff pre-Hilbert space [1] [2] is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets (as in ). [3] Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, [4] also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. [5]

Contents

An inner product naturally induces an associated norm, ( and are the norms of and in the picture), which canonically makes every inner product space into a normed vector space. If this normed space is also complete (i.e., a Banach space) then the inner product space is called a Hilbert space . [1] If an inner product space is not a Hilbert space then it can be "extended" to a Hilbert space called a completion. Explicitly, this means that is linearly and isometrically embedded onto a dense vector subspace of and that the inner product on is the unique continuous extension of the original inner product . [1] [6]

Definition

In this article, the field of scalars denoted is either the field of real numbers or the field of complex numbers .

Formally, an inner product space is a vector space over the field together with a map

called an inner product that satisfies the following conditions (1), (2), and (3) [1] for all vectors and all scalars : [7] [8] [9]

  1. Linearity in the first argument: [note 1]

     

     

     

     

    (Homogeneity in the 1st argument)

     

     

     

     

    (Additivity in the 1st argument)

    • If condition (1) holds and if is also antilinear (also called, conjugate linear) in its second argument [note 2] then is called a sesquilinear form . [1]
    • Each of these two properties imply for every vector [proof 1]
  2. Conjugate symmetry or Hermitian symmetry: [note 3]

     

     

     

     

    (Conjugate symmetry)

    • Conditions (1) and (2) are the defining properties of a Hermitian form , which is a special type of sesquilinear form. [1] A sesquilinear form is Hermitian if and only if is real for all [1] In particular, condition (2) implies [proof 2] that is a real number for all
  3. Positive definiteness : [1]

     

     

     

     

    (Positive definiteness)

The above three conditions are the defining properties of an inner product, which is why an inner product is sometimes (equivalently) defined as being a positive-definite Hermitian form. An inner product can equivalently be defined as a positive-definite sesquilinear form. [1] [note 4]

Assuming (1) holds, condition (3) will hold if and only if both conditions (4) and (5) below hold: [6] [1]

  1. Positive semi-definiteness or nonnegative-definiteness: [1]

     

     

     

     

    (Positive semi-definiteness)

    • Conditions (1), (2), and (4) are the defining properties of a positive semi-definite Hermitian form , which allows for the definition of a canonical seminorm on given by This seminorm is a norm if and only if condition (5) is satisfied.
  2. Point-separating or definiteness :

     

     

     

     

    (Point-separating)

Conditions (1) through (5) are satisfied by every inner product.

Elementary properties

Positive definiteness ensures that:

while is guaranteed by both homogeneity in the 1st argument and also by additivity in the 1st argument . [proof 1]

For every vector conjugate symmetry guarantees which implies that is a real number. It also guarantees that for all vectors and

where denotes the real part of a scalar

Conjugate symmetry and linearity in the first variable imply [proof 3] conjugate linearity, also known as antilinearity, in the second argument; explicitly, this means that for any vectors and any scalar

 

 

 

 

(Antilinearity in the 2nd argument)

This shows that every inner product is also a sesquilinear form and that inner products are additivity in each argument, meaning that for all vectors

Additivity in each argument implies the following important generalization of the familiar square expansion:

where

In the case of conjugate symmetry reduces to symmetry and so sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. That is, when then

 

 

 

 

(Symmetry)

and the binomial expansion becomes:

Alternative definitions, notations and remarks

A common special case of the inner product, the scalar product or dot product, is written with a centered dot

Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. In those disciplines, we would write the inner product as (the bra–ket notation of quantum mechanics), respectively (dot product as a case of the convention of forming the matrix product as the dot products of rows of with columns of ). Here, the kets and columns are identified with the vectors of and the bras and rows with the linear functionals (covectors) of the dual space with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, [10] taking to be conjugate linear in rather than A few instead find a middle ground by recognizing both and as distinct notations—differing only in which argument is conjugate linear.

There are various technical reasons why it is necessary to restrict the base field to and in the definition. Briefly, the base field has to contain an ordered subfield in order for non-negativity to make sense, [11] and therefore has to have characteristic equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally, any quadratically closed subfield of or will suffice for this purpose (for example, algebraic numbers, constructible numbers). However, in the cases where it is a proper subfield (that is, neither nor ), even finite-dimensional inner product spaces will fail to be metrically complete. In contrast, all finite-dimensional inner product spaces over or such as those used in quantum computation, are automatically metrically complete (and hence Hilbert spaces).

In some cases, one needs to consider non-negative semi-definite sesquilinear forms. This means that is only required to be non-negative. Treatment for these cases are illustrated below.

Some examples

Real and complex numbers

Among the simplest examples of inner product spaces are and The real numbers are a vector space over that becomes a real inner product space when endowed with standard multiplication as its real inner product: [4]

The complex numbers are a vector space over that becomes a complex inner product space when endowed with the complex inner product

Unlike with the real numbers, the assignment does not define a complex inner product on

Euclidean vector space

More generally, the real -space with the dot product is an inner product space, [4] an example of a Euclidean vector space.

where is the transpose of

Complex coordinate space

The general form of an inner product on is known as the Hermitian form and is given by

where is any Hermitian positive-definite matrix and is the conjugate transpose of For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.

Hilbert space

The article on Hilbert spaces 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 space which induces an incomplete metric is the space of continuous complex valued functions and on the interval The inner product is

This space is not complete; consider for example, for the interval [−1, 1] the sequence of continuous "step" functions, defined by:

This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function.

Random variables

For real random variables and the expected value of their product

is an inner product. [12] [13] [14] In this case, if and only if (that is, almost surely), where denotes the probability of the event. This definition of expectation as inner product can be extended to random vectors as well.

Real matrices

For real square matrices of the same size, with transpose as conjugation

is an inner product.

Vector spaces with forms

On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism ), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.

Basic results, terminology, and definitions

Norm

Every inner product space induces a norm, called its canonical norm, that is defined by [4]

With this norm, every inner product space becomes a normed vector space.

As for every normed vector space, an inner product space is a metric space, for the distance defined by

The axioms of the inner product guarantee that the map above forms a norm, which will have the following properties.

Homogeneity
For a vector and a scalar
Triangle inequality
For vectors
These two properties show that one has indeed a norm.
Cauchy–Schwarz inequality
For vectors

with equality if and only if and are linearly dependent. In the Russian mathematical literature, this inequality is also known as the Cauchy–Bunyakovsky inequality or the Cauchy–Bunyakovsky–Schwarz inequality.

Cosine similarity
When is a real number then the Cauchy–Schwarz inequality guarantees that lies in the domain of the inverse trigonometric function and so the (non oriented) angle between and can be defined as:

where

Polarization identity
The inner product can be retrieved from the norm by the polarization identity

which is a form of the law of cosines.

Orthogonality
Two vectors and are called orthogonal, written if their inner product is zero: This happens if and only if for all scalars [15] Moreover, for the scalar minimizes with value For a complex − but not real − inner product space a linear operator is identically if and only if for every [15]
Orthogonal complement
The orthogonal complement of a subset is the set of all vectors such that and are orthogonal for all ; that is, it is the set

This set is always a closed vector subspace of and if the closure of in is a vector subspace then

Pythagorean theorem
Whenever and then

The proof of the identity requires 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 in Euclidean geometry.

Parseval's identity
An induction on the Pythagorean theorem yields: if are orthogonal vectors (meaning that for distinct indices ) then
Parallelogram law
For all
The parallelogram law is, in fact, a necessary and sufficient condition for the existence of a inner product corresponding to a given norm.
Ptolemy's inequality
For all

Ptolemy's inequality is, in fact, a necessary and sufficient condition for the existence of a inner product corresponding to a given norm. In detail, Isaac Jacob Schoenberg proved in 1952 that, given any real, seminormed space, if its seminorm is ptolemaic, then the seminorm is the norm associated with an inner product. [16]

Real and complex parts of inner products

Suppose that is an inner product on (so it is antilinear in its second argument). The polarization identity shows that the real part of the inner product is

If is a real vector space then

and the imaginary part (also called the complex part) of is always 0.

Assume for the rest of this section that is a complex vector space. The polarization identity for complex vector spaces shows that

The map defined by for all satisfies the axioms of the inner product except that it is antilinear in its first, rather than its second, argument. The real part of both and are equal to but the inner products differ in their complex part:

The last equality is similar to the formula expressing a linear functional in terms of its real part.

Real vs. complex inner products

Let denote considered as a vector space over the real numbers rather than complex numbers. The real part of the complex inner product is the map which necessarily forms a real inner product on the real vector space Every inner product on a real vector space is a bilinear and symmetric map.

For example, if with inner product where is a vector space over the field then is a vector space over and is the dot product where is identified with the point (and similarly for ). Also, had been instead defined to be the symmetric map (rather than the usual conjugate symmetric map ) then its real part would not be the dot product; furthermore, without the complex conjugate, if but then so the assignment does not define a norm.

The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if then but the next example shows that the converse is in general not true. Given any the vector (which is the vector rotated by 90°) belongs to and so also belongs to (although scalar multiplication of by is not defined in it is still true that the vector in denoted by is an element of ). For the complex inner product, whereas for the real inner product the value is always

If has the inner product mentioned above, then the map defined by is a non-zero linear map (linear for both and ) that denotes rotation by in the plane. This map satisfies for all vectors where had this inner product been complex instead of real, then this would have been enough to conclude that this linear map is identically (i.e. that ), which rotation is certainly not. In contrast, for all non-zero the map satisfies

Orthonormal sequences

Let be a finite dimensional inner product space of dimension Recall that every basis of consists of exactly linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis is orthonormal if for every and for each index

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let be any inner product space. Then a collection

is a basis for if the subspace of generated by finite linear combinations of elements of is dense in (in the norm induced by the inner product). Say that is an orthonormal basis for if it is a basis and

if and for all

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

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

Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Any complete inner product space has an orthonormal basis.

The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).[ citation needed ]

Parseval's identity leads immediately to the following theorem:

Theorem. Let be a separable inner product space and an orthonormal basis of Then the map

is an isometric linear map 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 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 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 inner product. The mapping

is an isometric linear map with dense image.

Orthogonality of the sequence follows immediately from the fact that if 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 between inner product spaces and are of relevance:

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.

Generalizations

Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

Degenerate inner products

If is a vector space and a semi-definite sesquilinear form, then the function:

makes sense and satisfies all the properties of norm except that does not imply (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient The sesquilinear form factors through

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.

Nondegenerate conjugate symmetric forms

Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero there exists some such that though need not equal ; in other words, the induced map to the dual space is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism ) and thus hold more generally.

The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a covector with an vector, yielding a matrix (a scalar), while the outer product is the product of an vector with a covector, yielding an matrix. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.

As a further complication, in geometric algebra the inner product and the exterior (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the outer product (alternatively, wedge product). The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

See also

Notes

  1. By combining the linear in the first argument property with the conjugate symmetry property you get conjugate-linear in the second argument: This is how the inner product was originally defined and is still used in some old-school math communities. However, all of engineering and computer science, and most of physics and modern mathematics now define the inner product to be linear in the second argument and conjugate-linear in the first argument because this is more compatible with several other conventions in mathematics. Notably, for any inner product, there is some hermitian, positive-definite matrix such that (Here, is the conjugate transpose of )
  2. This means that and for all vectors and all scalars
  3. A bar over an expression denotes complex conjugation; for instance, is the complex conjugation of For real values, and conjugate symmetry is equivalent to symmetry .
  4. This is because condition (1) (i.e. linearity in the first argument) and positive definiteness implies that is always a real number. And as mentioned before, a sesquilinear form is Hermitian if and only if is real for all
Proofs
  1. 1 2 Homogeneity in the 1st argument implies Additivity in the 1st argument implies so adding to both sides proves
  2. A complex number is a real number if and only if Using in condition (2) gives which implies that is a real number.
  3. Let be vectors and let be a scalar. Then and

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".

Riesz representation theorem, sometimes called Riesz–Fréchet representation theorem, named after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next; a natural isomorphism.

The Cauchy–Schwarz inequality is considered to be one of the most important and widely used inequalities in all of mathematics.

Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, the complex conjugate of is equal to The complex conjugate of is often denoted as

In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In mathematics, a linear form is a linear map from a vector space to its field of scalars.

In mathematics, the conjugate transpose of an m-by-n matrix with complex entries is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry. It is often denoted as or .

In mathematics, a function between two real or complex vector spaces is said to be antilinear or conjugate-linear if

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint. Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number.

In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by . If the vectors are real and the columns of matrix , then the Gram matrix is .

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.

In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .

Polarization identity Formula relating the norm and the inner product in a inner product space

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Equivalently, the polarization identity describes when a norm can be assumed to arise from an inner product. In that terminology:

Convex analysis

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

Hilbert space Generalization of Euclidean space allowing infinite dimensions

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

In mathematics, a square-integrable function, also called a quadratically integrable function or function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows.

In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer. which is an inner product not required to be conjugate symmetric. It was formulated by Günter Lumer, for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis. Fundamental properties were later explored by Giles.

In the field of functional analysis, a subfield of mathematics, a dual system, dual pair, or a duality over a field is a triple consisting of two vector spaces over and a bilinear map such that for all non-zero the map is not identically and for all non-zero the map is not identically 0. The study of dual systems is called duality theory.

This is a glossary for the terminology in a mathematical field of functional analysis.

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