- Inner product space
In

mathematics , an**inner product space**is avector space with the additionalstructure of**inner product**. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as thelength of a vector or theangle between two vectors. It also provides the means of definingorthogonality between vectors (zero scalar product). Inner product spaces generalizeEuclidean space s (in which the inner product is thedot product , also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied infunctional 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 aHilbert 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 eitherthe field ofreal number s**R**or the field ofcomplex number s**C**. See below.Formally, an inner product space is a

vector space "V" over the field**F**together with apositive-definite sesquilinear form, called an "inner product". For real vector spaces, this is actually a positive-definite symmetricbilinear form . Thus the inner product is a

$langle\; cdot,\; cdot\; angle\; :\; V\; imes\; V\; ightarrow\; mathbb\{F\}$satisfying the following

axiom s for all $x,y,z\; in\; V,\; a,b\; in\; mathbb\{F\}$:* Conjugate symmetry:

::$langle\; x,y\; angle\; =overline\{langle\; y,x\; angle\}.$

:This condition implies that $langle\; x,x\; angle\; in\; mathbb\{R\}$, because $langle\; x,x\; angle\; =\; overline\{langle\; x,x\; angle\}$.

:"(Conjugation is also often written with an asterisk, as in $langle\; y,x\; angle^\{*\}$, as is the

conjugate transpose .)"*

Linear ity in the first variable:::$langle\; ax,y\; angle=\; a\; langle\; x,y\; angle.$::$langle\; x+y,z\; angle=\; langle\; x,z\; angle+\; langle\; y,z\; angle.$

:By combining these with conjugate symmetry, we get:::$langle\; x,by\; angle=\; overline\{b\}\; langle\; x,y\; angle.$::$langle\; x,y+z\; angle=\; langle\; x,y\; angle+\; langle\; x,z\; angle.$:So $langle\; cdot\; ,\; cdot\; angle$ is actually a

sesquilinear form.* Positivity:

::$langle\; x,x\; angle\; >\; 0$ for all $x\; e\; 0$.:"(This makes sense because $langle\; x,x\; angle\; in\; mathbb\{R\}$ for all $xin\; V$.)"

* Definiteness:::$langle\; x,x\; angle\; =\; 0$ implies $x\; =\; 0$.

:Hence, the inner product is a positive-definite

Hermitian form .:The property of an inner product space $V$ that::$langle\; x+y,z\; angle=\; langle\; x,z\; angle+\; langle\; y,z\; angle$ and $langle\; x,y+z\; angle\; =\; langle\; x,y\; angle\; +\; langle\; x,z\; angle$ is known as "additivity".

:Note that if

**F**=**R**, then the conjugate symmetry property is simply "symmetry" of the inner product, i.e.,:: $langle\; x,y\; angle=langle\; y,x\; angle.$: In this case, sesquilinearity becomes standardbilinearity .**Remark**: In the more abstract linear algebra literature, the conjugate-linear argument of the inner product is conventionally put in the second position (e.g., "y" in $langle\; x,y\; angle$) as we have done above. This convention is reversed in bothphysics andmatrix algebra ; in those respective disciplines we would write the product $langle\; x,y\; angle$ as $langle\; y|x\; angle$ (thebra-ket notation ofquantum mechanics ) and $y^Tx$ (dot product as a case of the convention of forming the matrix product "AB" as the dot products of rows of "A" with columns of "B"). Here the kets and columns are identified with the vectors of "V" and the bras and rows with thedual vector s orlinear functional s of thedual space "V"^{*}, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature, e.g., Emch [1972] , taking $langle\; x,y\; angle$ to be conjugate linear in "x" rather than "y". 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

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. More generally any quadratically closed subfield of**R**or**C**will suffice for this purpose, e.g., thealgebraic number s, but when it is a proper subfield (i.e., neither**R**nor**C**) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over**R**or**C**, such as those used inquantum computation , are automatically metrically complete and hence Hilbert spaces.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 is the

real numbers with the standard multiplication as the inner product:$langle\; x,y\; angle\; :=\; xy$More generally any Euclidean space

**R**^{"n"}with thedot product is an inner product space:$langle\; (x\_1,ldots,\; x\_n),(y\_1,ldots,\; y\_n)\; angle\; :=\; sum\_\{i=1\}^\{n\}\; x\_i\; y\_i\; =\; x\_1\; y\_1\; +\; cdots\; +\; x\_n\; y\_n$The general form of an inner product on

**C**^{"n"}is given by::$langle\; mathbf\{x\},mathbf\{y\}\; angle\; :=\; mathbf\{y\}^*mathbf\{M\}mathbf\{x\}$with " M" any symmetric

positive-definite matrix , and**y**^{*}theconjugate transpose of**y**. For the real case this corresponds to the dot product of the results of directionally differential scaling of the two vectors, with positivescale factor s 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:$langle\; f\; ,\; g\; angle\; :=\; int\_a^b\; f(t)\; overline\{g(t)\}\; ,\; dt$

This space is not complete; consider for example, for the interval [−1,1] the sequence of "step" functions { "f"

_{"k"}}_{"k"}where

* "f"_{"k"}("t") is 0 for "t" in the subinterval [−1,0]

* "f"_{"k"}("t") is 1 for "t" in the subinterval [1/"k", 1]

* "f"_{"k"}is affine in [0, 1/"k"] This sequence is aCauchy sequence which does not converge to a "continuous" function.**Norms on inner product spaces**Inner product spaces have a naturally defined norm

:$|x|\; =sqrt\{langle\; x,\; x\; angle\}.$

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":: $|langle\; x,y\; angle|\; leq\; |x|\; cdot\; |y|$

: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.

::It is trivial to prove the inequality true in the case "y" = 0. Thus we assume <"y", "y"> is nonzero, giving us the following:

::$lambda\; =\; langle\; y\; ,\; y\; angle^\{-1\}\; langle\; x,\; y\; angle$::$0\; leq\; langle\; x\; -lambda\; y,\; x\; -lambda\; y\; angle\; =\; langle\; x,\; x\; angle\; -\; langle\; y\; ,\; y\; angle^\{-1\}\; |\; langle\; x,y\; angle|^2.$

::The complete proof can be obtained by multiplying out this result.

*

Orthogonal ity: 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 theangle between two non-zero vectors "x" and "y" (at least in the case "F" =**R**) by the identity:$operatorname\{angle\}(x,y)\; =\; arccos\; frac\{langle\; x,\; y\; angle\}\{|x|\; cdot\; |y.$

: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-dimensionalEuclidean 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

::$|r\; cdot\; x|\; =\; |r|\; cdot\; |\; x|.$

:The homogeneity property is completely trivial to prove.

*

Triangle inequality : for "x", "y" elements of "V"::$|x\; +\; y|\; leq\; |x\; |\; +\; |y|.$

: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 ametric space . The most important inner product spaces are the ones which are complete with respect to this metric; they are calledHilbert space s. 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 : for "x", "y" elements of "V",::$|x\; +\; y|^2\; +\; |x\; -\; y|^2\; =\; 2|x|^2\; +\; 2|y|^2.$

*

Pythagorean theorem : Whenever "x", "y" are in "V" and <"x", "y"> = 0, then::$|x|^2\; +\; |y|^2\; =\; |x+y|^2.$

: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. Alternatively, both can be seen as consequences of the identity

::$|x\; +\; y|^2\; =\; |x|^2\; +\; |y|^2\; +\; 2\; eal\; langle\; x\; ,\; y\; angle.$

:which is a form of the

law of cosines , and is proved as before.: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 "x"

_{1}, ..., "x"_{"n"}areorthogonal vectors, that is, <"x"_{"j"}, "x"_{"k"}> = 0 for distinct indices "j", "k", then::$sum\_\{i=1\}^n\; |x\_i|^2\; =\; left|sum\_\{i=1\}^n\; x\_i\; ight|^2.$

: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::$sum\_\{i=1\}^infty|x\_i|^2\; =\; left|sum\_\{i=1\}^infty\; x\_i\; ight|^2,$

:"provided the infinite series on the left is

convergent ." Completeness of the space is needed to ensure that the sequence of partial sums::$S\_k\; =\; sum\_\{i=1\}^k\; x\_i$

: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 ofdense 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":$operatorname\{span\}\{v\_1,\; ldots,\; v\_n\}\; =\; operatorname\{span\}\{e\_1,\; ldots,\; e\_n\}$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

$x\; mapsto\; \{langle\; e\_k,\; x\; angle\}\_\{k\; in\; mathbb\{N$is an isometric linear map "V" → "l"^{2}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 oftrigonometric polynomial s. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided "l^{2}" is defined appropriately, as is explained in the articleHilbert space ).In particular, we obtain the following result in the theory of Fourier series:**Theorem**. Let "V" be the inner product space $C\; [-pi,pi]$. Then the sequence (indexed on set of all integers) of continuous functions:$e\_k(t)\; =\; (2\; pi)^\{-1/2\}e^\{i\; k\; t\}$is an orthonormal basis of the space $C\; [-pi,pi]$ with the "L"^{2}inner product. The mapping:$f\; mapsto\; frac\{1\}\{sqrt\{2\; pi\; left\{int\_\{-pi\}^pi\; f(t)\; e^\{-i\; k\; t\}\; ,\; dt\; ight\}\_\{k\; in\; mathbb\{Z$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 :$int\_\{-pi\}^pi\; e^\{-i\; (j-k)\; t\}\; ,\; dt\; =\; 0.$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 $[-pi,pi]$ 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 , 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 areinjective . Isometries aremorphism s between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare withorthogonal matrix ).

* Isometrical isomorphisms, i.e. "A" is an isometry which issurjective (and hencebijective ). Isometrical isomorphisms are also known as unitary operators (compare withunitary 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 generallynormal operator s 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 asemi-norm .) We can produce an inner product space by considering thequotient "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 ofsemi-definite kernel s on arbitrary sets.**ee also***

Outer product

*Exterior algebra

*bilinear form

*dual space

*dual pair

*biorthogonal system

*Fubini-Study metric

*Energetic space **References***

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