Inner product space

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In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space[1][2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used 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.[3]

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, (denoted and in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space.[1] If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space This means that is a linear subspace of the inner product of is the restriction of that of and is dense in for the topology defined by the norm.[1][4]

Definition edit

In this article, F denotes a field that is either the real numbers   or the complex numbers   A scalar is thus an element of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted   for distinguishing it from the scalar 0.

An inner product space is a vector space V over the field F together with an inner product, that is, a map


that satisfies the following three properties for all vectors   and all scalars  .[5][6]

  • Conjugate symmetry:
    As   if and only if   is real, conjugate symmetry implies that   is always a real number. If F is  , conjugate symmetry is just symmetry.
  • Linearity in the first argument:[Note 1]
  • Positive-definiteness: if   is not zero, then
    (conjugate symmetry implies that   is real).

If the positive-definiteness condition is replaced by merely requiring that   for all  , then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form   is an inner product if and only if for all  , if   then  .[7]

Basic properties edit

In the following properties, which result almost immediately from the definition of an inner product, x, y and z are arbitrary vectors, and a and b are arbitrary scalars.

  •   is real and nonnegative.
  •   if and only if  
    This implies that an inner product is a sesquilinear form.
  •   where  
    denotes the real part of its argument.

Over  , conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a positive-definite symmetric bilinear form. The binomial expansion of a square becomes


Convention variant edit

Some authors, especially in physics and matrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. Bra-ket notation in quantum mechanics also uses slightly different notation, i.e.  , where  .

Notation edit

Several notations are used for inner products, including  ,  ,   and  , as well as the usual dot product.

Some examples edit

Real and complex numbers edit

Among the simplest examples of inner product spaces are   and   The real numbers   are a vector space over   that becomes an inner product space with arithmetic multiplication as its inner product:


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

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

Euclidean vector space edit

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

where   is the transpose of  

A function   is an inner product on   if and only if there exists a symmetric positive-definite matrix   such that   for all   If   is the identity matrix then   is the dot product. For another example, if   and   is positive-definite (which happens if and only if   and one/both diagonal elements are positive) then for any  

As mentioned earlier, every inner product on   is of this form (where   and   satisfy  ).

Complex coordinate space edit

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 edit

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 edit

For real random variables   and   the expected value of their product

is an inner product.[8][9][10] 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.

Complex matrices edit

The inner product for complex square matrices of the same size is the Frobenius inner product  . Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by,

Finally, since for   nonzero,  , we get that the Frobenius inner product is positive definite too, and so is an inner product.

Vector spaces with forms edit

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 edit

Norm properties edit

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

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

So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties:

Absolute homogeneity
for every   and   (this results from  ).
Triangle inequality
for   These two properties show that one has indeed a norm.
Cauchy–Schwarz inequality
for every   with equality if and only if   and   are linearly dependent.
Parallelogram law
for every   The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.
Polarization identity
for every   The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of  
Ptolemy's inequality
for every   Ptolemy's inequality is a necessary and sufficient condition for a seminorm to be the norm defined by an inner product.[11]

Orthogonality edit

Two vectors   and   are said to be orthogonal, often written   if their inner product is zero, that is, if  
This happens if and only if   for all scalars  [12] and if and only if the real-valued function   is non-negative. (This is a consequence of the fact that, if   then the scalar   minimizes   with value   which is always non positive).
For a complex inner product space   a linear operator   is identically   if and only if   for every  [12] This is not true in general for real inner product spaces, as it is a consequence of conjugate symmetry being distinct from symmetry for complex inner products. A counterexample in a real inner product space is   a 90° rotation in  , which maps every vector to an orthogonal vector but is not identically  .
Orthogonal complement
The orthogonal complement of a subset   is the set   of the vectors that are orthogonal to all elements of C; that is,
This set   is always a closed vector subspace of   and if the closure   of   in   is a vector subspace then  
Pythagorean theorem
If   and   are orthogonal, then
This may be proved by expressing the squared norms in terms of the inner products, using additivity for expanding the right-hand side of the equation.
The name Pythagorean theorem arises from the geometric interpretation in Euclidean geometry.
Parseval's identity
An induction on the Pythagorean theorem yields: if   are pairwise orthogonal, then
When   is a real number then the Cauchy–Schwarz inequality implies that   and thus that
is a real number. This allows defining the (non oriented) angle of two vectors in modern definitions of Euclidean geometry in terms of linear algebra. This is also used in data analysis, under the name "cosine similarity", for comparing two vectors of data.

Real and complex parts of inner products edit

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  

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.

These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on   considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space   and real inner products on  

For example, suppose that   for some integer   When   is considered as a real vector space in the usual way (meaning that it is identified with the  dimensional real vector space   with each   identified with  ), then the dot product   defines a real inner product on this space. The unique complex inner product   on   induced by the dot product is the map that sends   to   (because the real part of this map   is equal to the dot product).

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  ); thus the standard inner product   on   is an "extension" the dot product . 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   would 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   the vector in   denoted by   is nevertheless still also an element of  ). For the complex inner product,   whereas for the real inner product the value is always  

If   is a complex inner product and   is a continuous linear operator that satisfies   for all   then   This statement is no longer true if   is instead a real inner product, as this next example shows. Suppose that   has the inner product   mentioned above. Then the map   defined by   is a linear map (linear for both   and  ) that denotes rotation by   in the plane. Because   and   are perpendicular vectors and   is just the dot product,   for all vectors   nevertheless, this rotation map   is certainly not identically   In contrast, using the complex inner product gives   which (as expected) is not identically zero.

Orthonormal sequences edit

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 edit

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:   satisfies   for all   A linear isometry (resp. an antilinear isometry) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the polarization identity can be used to show that   is an isometry if and only if   for all   All isometries are injective. The Mazur–Ulam theorem establishes that every surjective isometry between two real normed spaces is an affine transformation. Consequently, an isometry   between real inner product spaces is a linear map if and only if   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.[13]

Generalizations edit

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 edit

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 edit

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 products edit

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. 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 edit

Notes edit

  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 used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the bra-ket notation of Paul Dirac, where the inner product is taken to be linear in the second argument and conjugate-linear in the first argument; this convention is used in many other domains such as engineering and computer science.

References edit

  1. ^ a b c Trèves 2006, pp. 112–125.
  2. ^ Schaefer & Wolff 1999, pp. 40–45.
  3. ^ Moore, Gregory H. (1995). "The axiomatization of linear algebra: 1875-1940". Historia Mathematica. 22 (3): 262–303. doi:10.1006/hmat.1995.1025.
  4. ^ Schaefer & Wolff 1999, pp. 36–72.
  5. ^ 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.
  6. ^ Prugovečki, Eduard (1981). "Definition 2.1". Quantum Mechanics in Hilbert Space (2nd ed.). Academic Press. pp. 18ff. ISBN 0-12-566060-X.
  7. ^ Schaefer & Wolff 1999, p. 44.
  8. ^ Ouwehand, Peter (November 2010). "Spaces of Random Variables" (PDF). AIMS. Archived from the original (PDF) on 2017-09-05. Retrieved 2017-09-05.
  9. ^ Siegrist, Kyle (1997). "Vector Spaces of Random Variables". Random: Probability, Mathematical Statistics, Stochastic Processes. Retrieved 2017-09-05.
  10. ^ 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.
  11. ^ Apostol, Tom M. (1967). "Ptolemy's Inequality and the Chordal Metric". Mathematics Magazine. 40 (5): 233–235. doi:10.2307/2688275. JSTOR 2688275.
  12. ^ a b Rudin 1991, pp. 306–312.
  13. ^ Rudin 1991

Bibliography edit