Inner product space

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]

Geometric interpretation of the angle between two vectors defined using an inner product
Scalar product spaces, inner product spaces, Hermitian product spaces.
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.

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 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]

DefinitionEdit

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 propertiesEdit

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 remarksEdit

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 examplesEdit

Real and complex numbersEdit

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 spaceEdit

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 spaceEdit

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 spaceEdit

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 variablesEdit

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 matricesEdit

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

 
is an inner product.

Vector spaces with formsEdit

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 definitionsEdit

NormEdit

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] 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 productsEdit

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 sequencesEdit

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 spacesEdit

Several types of linear maps   between inner product spaces   and   are of relevance:

  • Continuous linear maps:   is linear and continuous with respect to the metric defined above, or equivalently,   is linear and the set of non-negative reals   where   ranges over the closed unit ball of   is bounded.
  • Symmetric linear operators:   is linear and   for all  
  • Isometries:   is linear and   for all   or equivalently,   is linear and   for all   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:   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.

GeneralizationsEdit

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 productsEdit

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 formsEdit

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.

Related productsEdit

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 alsoEdit

NotesEdit

  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. ^ a b 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    

ReferencesEdit

  1. ^ a b c d e f g h i j k Trèves 2006, pp. 112-125.
  2. ^ Schaefer & Wolff 1999, pp. 40-45.
  3. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-25.
  4. ^ a b c d Weisstein, Eric W. "Inner Product". mathworld.wolfram.com. Retrieved 2020-08-25.
  5. ^ Moore, Gregory H. (1995). "The axiomatization of linear algebra: 1875-1940". Historia Mathematica. 22 (3): 262–303. doi:10.1006/hmat.1995.1025.
  6. ^ a b Schaefer & Wolff 1999, pp. 36-72.
  7. ^ Jain, P. K.; Ahmad, Khalil (1995). "5.1 Definitions and basic properties of inner product spaces and Hilbert spaces". Functional Analysis (2nd ed.). New Age International. p. 203. ISBN 81-224-0801-X.
  8. ^ Prugovec̆ki, Eduard (1981). "Definition 2.1". Quantum Mechanics in Hilbert Space (2nd ed.). Academic Press. pp. 18ff. ISBN 0-12-566060-X.
  9. ^ "Inner Product Space | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-25.
  10. ^ Emch, Gerard G. (1972). Algebraic Methods in Statistical Mechanics and Quantum Field Theory. New York: Wiley-Interscience. ISBN 978-0-471-23900-0.
  11. ^ Finkbeiner, Daniel T. (2013), Introduction to Matrices and Linear Transformations, Dover Books on Mathematics (3rd ed.), Courier Dover Publications, p. 242, ISBN 9780486279664.
  12. ^ Ouwehand, Peter (November 2010). "Spaces of Random Variables" (PDF). AIMS. Retrieved 2017-09-05.
  13. ^ Siegrist, Kyle (1997). "Vector Spaces of Random Variables". Random: Probability, Mathematical Statistics, Stochastic Processes. Retrieved 2017-09-05.
  14. ^ Bigoni, Daniele (2015). "Appendix B: Probability theory and functional spaces" (PDF). Uncertainty Quantification with Applications to Engineering Problems (PhD). Technical University of Denmark. Retrieved 2017-09-05.
  15. ^ a b Rudin 1991, pp. 306-312.
  16. ^ Apostol, Tom M. (1967). "Ptolemy's Inequality and the Chordal Metric". Mathematics Magazine. 40 (5): 233–235. doi:10.2307/2688275. JSTOR 2688275.

BibliographyEdit