Inner Product Space

#REDIRECT Inner product space

Inner product space

In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. Inner product spaces are generalizations of Euclidean space (with the dot product as the inner product) and are studied in functional analysis. An inner product space is also called a ''pre-Hilbert space'', since its Complete space#Completion with respect to the Metric space#Formal definition Normed vector space#Distances in normed vector spaces by its inner product is a Hilbert space. Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used. == Definitions == In the following article, the field (mathematics) of scalars denoted F is either the field of real numbers R or the field of complex numbers C. See below. Formally, an inner product space is a vector space ''V'' over the field F together with a map, called an ''inner product'' :

\langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbf{F}

satisfying the following axioms: * Additivity: ::

\forall x,y,z \in V,\ \langle x,y+z\rangle = \langle x,y \rangle + \langle x,z \rangle

:: and

\langle x+y,z\rangle = \langle x,z \rangle + \langle y,z \rangle

* Nonnegativity: ::

\forall x \in V,\ \langle x,x\rangle \ge 0.

* Nondegeneracy: ::

\forall x \in V,\ \langle x,x\rangle = 0 \mbox{ iff } x = 0.

* complex conjugate symmetry: ::

\forall x,y\in V,\ \langle x,y\rangle =\overline{\langle y,x\rangle}

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

\forall b\in F,\ \forall x,y\in V,\ \langle x,by\rangle= b \langle x,y\rangle

\forall x,y,z\in V,\ \langle x,y+z\rangle= \langle x,y\rangle+ \langle x,z\rangle.

:By combining these with conjugate symmetry, we get: ::

\forall a\in F,\ \forall x,y\in V,\ \langle ax,y\rangle= \overline{a} \langle x,y\rangle

\forall x,y,z\in V,\ \langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.

Note that if F=R, then the conjugate symmetry property is simply ''symmetry'' of the inner product, i.e. ::

\langle x,y\rangle=\langle y,x\rangle

In this case, sesquilinearity becomes standard linearity. Remark. Many mathematical authors require an inner product to be linear in the first argument and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the bra-ket notation used by physicists in quantum mechanics and is now often used by mathematicians as well. Some authors adopt the convention that < , > is linear in the first component while < | > is linear in the second component, although this is by no means universal. For instance the G. Emch reference does not follow this convention. There are various technical reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered field (in order for non-negativity to make sense) and therefore has to have characteristic (algebra) equal to 0. This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. In some cases we need to consider non-negative ''semi-definite'' sesquilinear forms. This means that <''x'', ''x''> is only required to be non-negative. We show how to treat these below. == Examples == A trivial example are the real numbers with the standard multiplication as the inner product :

\langle x,y\rangle := xy

More generally any Euclidean space R^''n'' with the dot product is an inner product space :

\langle (x_1,\ldots, x_n),(y_1,\ldots, y_n)\rangle := \sum_{i=1}^{n} x_i y_i = x_1 y_1 + \cdots + x_n y_n

Even more generally any positive-definite matrix ''M'' can be used to define an inner product on C^''n'' as :

\langle \mathbf{x},\mathbf{y}\rangle := \mathbf{x}^*\mathbf{M}\mathbf{y}

with x^* the conjugate transpose of x. 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 \rangle := \int_a^b \overline{f(t)} g(t) \, dt

This space is not complete; consider for example, for the interval [0,1] the sequence of functions { ''f''_''k'' }_''k'' where * ''f''_''k''(''t'') is 1 for ''t'' in the subinterval [0, 1/2] * ''f''_''k''(''t'') is 0 for t in the subinterval [1/2 + 1/''k'', 1] * ''f''_''k'' is affine in [1/2, 1/2 + 1/''k''] This sequence is a Cauchy sequence which does not converge to a ''continuous'' function. == Norms on inner product spaces == Inner product spaces have a naturally defined Normed vector space :

\|x\| =\sqrt{\langle x, x\rangle}.

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\rangle| \leq \|x\| \cdot \|y\|

:with equality if and only if ''x'' and ''y'' are linearly independent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the ''Cauchy-Bunyakowski-Schwarz inequality''. :Because of its importance, its short proof should be noted. To prove this inequality note it is trivial in the case ''y'' = 0. Thus we may assume <''y'', ''y''> is nonzero. Thus we may let ::

\lambda = \langle y , y \rangle^{-1} \langle y, x\rangle

:and it follows that ::

0 \leq \langle x -\lambda y,  x -\lambda y \rangle = \langle x, x\rangle - \langle y , y \rangle^{-1} | \langle x,y\rangle|^2.

:multiplying out, the result follows. :The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy-Schwarz inequality is that it justifies defining the angle between two non-zero vectors ''x'' and ''y'' (at least in the case ''F'' = R) by the identity ;:

\operatorname{angle}(x,y) = \arccos \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}.

:We assume the value of the angle is chosen to be in the interval (−π, +π]. This is in analogy to the familiar situation in two-dimensional Euclidean space. Correspondingly, we will say that non-zero vectors ''x'', ''y'' of ''V'' are orthogonal iff their inner product is zero. *Homogeneity: for ''x'' an element of ''V'' an ''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 a metric space. The most important inner product spaces are the ones which are completeness (topology) with respect to this metric; they are called Hilbert spaces. Every inner product ''V'' space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by ''V'' and is constructed by completing ''V''. *Parallelogram law: ::

\|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. 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 mathematical induction on the Pythagorean theorem yields: *If ''x''₁, ..., ''x''_''n'' are orthogonal 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 \right\|^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\right\|^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'' iff it is orthogonal and each ''e_k'' has norm 1. An ''orthonormal basis'' for an inner product space ''V'' is an orthonormal sequence whose algebraic span is ''V''. 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 map :

x \mapsto \{\langle e_k, x\rangle\}_{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 of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided ''l²'' is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series: Theorem. Let ''V'' be the inner product space

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 \right\}_{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 {||''Ax''||}, where ''x'' ranges over the closed unit ball of ''V'', is bounded. * Symmetric linear operators, i.e. ''A'' is linear and <''Ax'', ''y''> = <''x'', ''A y''> for all ''x'', ''y'' in ''V''. * Isometries, i.e. ''A'' is linear and <''Ax'', ''Ay''> = <''x'', ''y''> for all ''x'', ''y'' in ''V'', or equivalently, ''A'' is linear and ||''Ax''|| = ||''x''|| for all ''x'' in ''V''. All isometries are injective. * Isometrical isomorphisms, i.e. ''A'' is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix). From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces. == Degenerate inner products == If ''V'' is a vector space and < , > a semi-definite sesquilinear form, then the function ||''x''|| = <''x'',''x''>^1/2 makes sense and satisfies all the properties of norm except that ||''x''|| = 0 does not imply ''x'' = 0. We can produce an inner product space by considering the quotient ''W'' = ''V''/{''x'':||''x''|| = 0}. The sesquilinear form < , > factors through ''W''. This construction is used in numerous contexts. The Gelfand-Naimark-Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets. == References == * S. Axler, ''Linear Algebra Done Right'', Springer, 2004 * G. Emch, ''Algebraic Methods in Statistical Mechanics and Quantum Field Theory'', Wiley Interscience, 1972. * N. Young, ''An Introduction to Hilbert Spaces'', Cambridge University Press, 1988 ==See also== * outer product * bilinear form and links therein Functional analysis Norm

Inner product space

This definition is too narrow. Inner products may be negative or zero. The definition given is for a positive definite inner product. :That is a very context-dependent point. In many contexts, ''inner product'' is indeed taken to mean a positive-definite bilinear form. User:Michael Hardy 02:17, 8 Oct 2004 (UTC) To illustrate that, and how, the Triangle inequality is obtained as a consequence of the Cauchy-Schwarz inequality: Priliminaries, directly from the axioms: : <1, 1> > 0. : <( - ), ( - )> >= 0, : <( - ), ( - )> = : ( - ) <1, ( - )> = : -( - ) <1, ( - )> = : -( - ) <1, 1> ( - ) = : -( - )^2 <1, 1> >= 0; therefore ( - )^2 =< 0. Now adding the Cauchy-Bunyakovski-Schwarz inequality: 4 =< 4 : ( - )^2 + 4 =< 4 , ( + )^2 =< 4 . According to preliminaries also: 0 =< ( + )^2. Together with axioms, therefore ( + ) =< 2 Sqrt( ), ( + ) + + =< 2 Sqrt( ) + + , <(x + y), (x + y)> =< (Sqrt( ) + Sqrt( ))^2 Finally, per axioms 0 =< <(x + y), (x + y)>; thereby Sqrt( <(x + y), (x + y)> ) =< Sqrt( ) + Sqrt( ), i.e. || x + y || =< || x || + || y ||. Regards, User:Fwappler 02:13 Mar 3, 2003 (UTC). == Sequilinearity == Though I am used to Sequilinearity as measing linear in 1st argument, most physicists use the other convention. I have edited some articles with this assumption; moreover, many of the quantum mechanics articles naturally follow the physicist's convention. At this point the physicist's convention seems the one to follow in Wikipedia.User:CSTAR 14:07, 11 Aug 2004 (UTC) :Agreed. There's some confusion, but I think the physicist's convention is the one to follow, since it has a good reason for using that order. The convention seems to be working its way into recent mathematics, as well. -User:FunnyMan3595 18:28, Oct 23, 2004 (UTC) == Bilinear? == From the definitions of inner product and bilinear operator, follows that the inner product over a complex vector space isn't a bilinear form (but it is for real vector spaces). So, the definition in the article contains a wrong generalization: "Formally, an inner product space is a vector space V over the field F together with a bilinear form, called an inner product" : Good point. We could split it into two cases, or we could rewrite sesquilinear form to also apply to R. -- User:Waltpohl 07:25, 12 Jan 2005 (UTC) : Fixed. -- User:Waltpohl 01:00, 13 Jan 2005 (UTC) :: I think the inner product map should be ''V'' × ''V'' → F and not ''V'' × ''V'' → R. At least the standard inner product on C is not real-valued. I changed the article accordingly. -- User:Jitse Niesen 12:00, 13 Jan 2005 (UTC) ::: Oops, you're right. -- User:Waltpohl 03:37, 14 Jan 2005 (UTC) I have never seen a definition of inner product which isn't linear in the first argument. I am a mathematician and consider the current version incorrect. == Connnection to Bra-Ket Notation == Could we add an explanation on how the inner-product is related to Dirac notation? Perhaps I am requesting a discussion on the notion of dual spaces and inner-products. In "Principles of Quantum Mechanics," Shankar mentions that one can think of the inner-product as a mapping from ''V*'' × ''V'' rather than a mapping from ''V'' × ''V'' where ''V*'' is the dual space of bras (obtained by taking the conjugate transpose of the kets). Also, it seems that we should mention another common notation for inner-products: (x,y). In fact, the page on bra-ket notation refers to the inner-product in this manner. (User:Droova forgot to sign) : You can surely add one more section to the article explaining the connection to bra-ket notation. User:Oleg Alexandrov 15:30, 5 Jun 2005 (UTC)

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